A Glimpse of Convexity and Helly-Type Results
A Glimpse of Convexity and Helly-Type
Results
Ljn
Shanghai Jiao Tong University, Shanghai, China
) [email protected]
April 10, 2008
A Glimpse of Convexity and Helly-Type Results
The book contains a wealth of material not usually found in
textbooks on linear algebra. Besides the already mentioned
results derived by means of calculus, one finds Carathéodory’s
theorem on extreme points and Helly’s theorem on the
intersection of convex sets, the Farkas-Minkowski theorem on
linear inequalities, von Neumann’s minimax theorem of game
theory, the theorems of Perron and Frobenius on matrices with
nonnegative entries, etc. – Review of the book: Peter D. Lax,
Linear Algebra, Wiley, 1997.
A Glimpse of Convexity and Helly-Type Results
The book contains a wealth of material not usually found in
textbooks on linear algebra. Besides the already mentioned
results derived by means of calculus, one finds Carathéodory’s
theorem on extreme points and Helly’s theorem on the
intersection of convex sets, the Farkas-Minkowski theorem on
linear inequalities, von Neumann’s minimax theorem of game
theory, the theorems of Perron and Frobenius on matrices with
nonnegative entries, etc. – Review of the book: Peter D. Lax,
Linear Algebra, Wiley, 1997.
We try to give you some ideas of the above interesting topics
in a series of talks. This first talk in centered around
A Glimpse of Convexity and Helly-Type Results
The book contains a wealth of material not usually found in
textbooks on linear algebra. Besides the already mentioned
results derived by means of calculus, one finds Carathéodory’s
theorem on extreme points and Helly’s theorem on the
intersection of convex sets, the Farkas-Minkowski theorem on
linear inequalities, von Neumann’s minimax theorem of game
theory, the theorems of Perron and Frobenius on matrices with
nonnegative entries, etc. – Review of the book: Peter D. Lax,
Linear Algebra, Wiley, 1997.
We try to give you some ideas of the above interesting topics
in a series of talks. This first talk in centered around
`convexity and Helly-type results.`
A Glimpse of Convexity and Helly-Type Results
Minkowski is generally credited with the first
systematic study of convex sets, and the introduction
of fundamental concepts such as supporting
hyperplanes and the supporting hyperplane theorem,
the Minkowski distance function, extreme points of a
convex set, and many others. – S. Boyd, L.
Vandenberghe, Convex Optimization, Cambridge
University Press, 2004.
A Glimpse of Convexity and Helly-Type Results
I found, upon inspection, that the Calculus of
Variations was too ‘messy’ for my taste, though the
geometric simplicity of the notion of convexity was
very appealing. This led to the thought that ‘If only
I can prove enough theorems having to do with
convexity per se, I won’t have to work in the
Calculus of Variations.’ – Victor L. Klee,
A Glimpse of Convexity and Helly-Type Results
I found, upon inspection, that the Calculus of
Variations was too ‘messy’ for my taste, though the
geometric simplicity of the notion of convexity was
very appealing. This led to the thought that ‘If only
I can prove enough theorems having to do with
convexity per se, I won’t have to work in the
Calculus of Variations.’ – Victor L. Klee, a
mathematician specialising in convex sets, functional
analysis, analysis of algorithms, optimization, and
combinatorics.
A Glimpse of Convexity and Helly-Type Results
Victor L. Klee (1925 – August 17, 2007)
A Glimpse of Convexity and Helly-Type Results
Klee’s Art Gallery Problem: Given a simple n-gon, what is the
minimum number of vertices from which it is possible to view
every point in the interior of the polygon?
A Glimpse of Convexity and Helly-Type Results
Klee’s Art Gallery Problem: Given a simple n-gon, what is the
minimum number of vertices from which it is possible to view
every point in the interior of the polygon?
Answer (Chvatal’s Art Gallery Theorem): bn/3c.
A Glimpse of Convexity and Helly-Type Results
ùìƒp?Òà+¶ìƒ$CY?§Ò]¬"
ù‘à’‘]’i§{5^<"X8†^ŠZ,ƒ¶§
ú#m§ØáœÏ"Œùü?˜þ˜e§˜²˜V§
˜p˜L§˜ì˜Y§¾´AÏd?"
– ùȎ, ù¢‰§1ԛ8£, à+,¬(a՘ ]¬
,é˜N("
A Glimpse of Convexity and Helly-Type Results
M.C. Escher: Concave and Convex
A Glimpse of Convexity and Helly-Type Results
Local vs Global, Geometry vs Algebra, Person vs Mathematics,
Duality (Symmetry)
The mathematical sciences particularly exhibit
order, symmetry, and limitation; and these are the
greatest forms of the beautiful. – Aristotle (384 –
322 BC)
A Glimpse of Convexity and Helly-Type Results
From Local Views to Global Ones
Understanding the interconnection between local and global
properties of mathematical objects are important in almost all
areas of mathematics.
In the classical period people on the whole would
have studied things on a small scale, in local coordinates
and so on. In this century, the emphasis has shifted to
try and understand the global, large-scale behavior. –
Mathematics in the 20th Century, – Sir Michael Atiyah,
Mathematics in the 20th century, Bulletin of the London
Mathematical Society 34 (2002), 1–15.
‡©AÛ̇¯K´N5§=ïĘm½6
/‡5Ÿ§cÙ´ÛÜ5Ÿ†N5Ÿ'
X"Gauss-BonnetúªÒ´˜‡~f"¨Ž
A Glimpse of Convexity and Helly-Type Results
Global Phenomena and Global Methods
N. Linial, Local-global phenomena in graphs, Combinatorics
Probability and Computing, 2 (1993), 491 – 503.
By “global” we mean those based on morphisms, i.e. maps
between instances of a problem which preserve the essential
features of that problem. This approach has been
systematically developed in algebra, ... Notions of symmetry,
product decomposition and reduction abound in the
combinatorial literature and these are by nature global
concepts. – L.H. Harper, Global Methods for Combinatorial
Isoperimetric Problems, Cambridge University Press, 2004.
A Glimpse of Convexity and Helly-Type Results
A Glimpse of Convexity and Helly-Type Results
Outline
1 Basics
2 Variations
3 Separating hyperplane and theorems of alternatives
A Glimpse of Convexity and Helly-Type Results
Basics
1˜Ü©
A Glimpse of Convexity and Helly-Type Results
Basics
Convexity
A set A ⊆ Rd is convex provided that for any two points
x, y ∈ A the line segment connecting x and y also lies in A.
A Glimpse of Convexity and Helly-Type Results
Basics
Convexity
A set A ⊆ Rd is convex provided that for any two points
x, y ∈ A the line segment connecting x and y also lies in A.
A function f : Rd → R is convex provided that for each
λ ∈ R the set {x ∈ Rd : f (x) ≤ λ} is convex.
A Glimpse of Convexity and Helly-Type Results
Basics
Convexity
A set A ⊆ Rd is convex provided that for any two points
x, y ∈ A the line segment connecting x and y also lies in A.
A function f : Rd → R is convex provided that for each
λ ∈ R the set {x ∈ Rd : f (x) ≤ λ} is convex.
You may be more familiar with convex function. But, do you
already notice that in mathematics and in many other
situations good things arrive in pairs?
A Glimpse of Convexity and Helly-Type Results
Basics
Convexity
A set A ⊆ Rd is convex provided that for any two points
x, y ∈ A the line segment connecting x and y also lies in A.
A function f : Rd → R is convex provided that for each
λ ∈ R the set {x ∈ Rd : f (x) ≤ λ} is convex.
You may be more familiar with convex function. But, do you
already notice that in mathematics and in many other
situations good things arrive in pairs?
Indeed, convexity has an immensely rich structure
and numerous applications. On the other hand,
almost every “convex” idea can be explained by a
two-dimensional picture. – A. Barvinok, A Course in
Convexity, AMS, 2002.
A Glimpse of Convexity and Helly-Type Results
Basics
Are They Convex?
1
2
Let Hn be the unit hypercube in Rn . Let A and B be
disjoint sets whose union is the set of 2n vertices of Hn .
Suppose that for any x, y ∈ A and x 0 , y 0 ∈ B we have
x + y 6= x 0 + y 0 . Is Hn \ Conv (A) a convex set?
Let q1 , q2 : Rn → R be quadratic forms and let S n−1 be
the unit sphere in Rn . Consider the map
T : Rn → R2 , T (x) = (q1 (x), q2 (x)). Is the image
T (S n−1 ) of the sphere convex?
A Glimpse of Convexity and Helly-Type Results
Basics
Helly’s Theorem
For any positive integer n, we denote by [n] the set of first n
positive integers, namely [n] = {1, 2, . . . , n}.
Theorem 1 (Helly’s Theorem)
Let Ci , i ∈ [n] be convex sets in Rd , n ≥ d + 1. Suppose that
the intersection of every d + 1 of these sets is nonempty. Then
∩i∈[n] Ci 6= ∅.
E. Helly, Ueber Mengen konvexer Koerper mit
gememschaftliches Punkten, Jber. DMV 32 (1923) 175-176.
Helly’s Theorem is in the same spirit with many compactness
results, which assert that some property holds for the whole
space if and only if it holds for every finite subset of the space.
A Glimpse of Convexity and Helly-Type Results
Basics
Helly, Hahn-Banach, and Tibor Rado
Eduard Helly (June 1, 1884 – Nov. 28, 1943) proved the
Hahn-Banach theorem in 1912, fifteen years before Hahn published
essentially the same proof and 20 years before Banach gave his
new setting. His famous Helly’s Theorem is also closely related to
his Hahn-Banach theorem.
Terence Tao, The Hahn-Banach theorem, Menger’s theorem, and
Helly’s theorem, an expository note,
http://www.math.ucla.edu/~tao/preprints/misc.html
After being shot, Helly had been captured by the Russians towards
the end of 1915 and after a spell in hospital was by then in the
same prison camp Radó. ... In the prison camp Helly acted as
mathematics teacher to Radó who was also able to read books on
mathematics. – http://www-groups.dcs.st-and.ac.uk/
~history/Biographies/Rado.html
A Glimpse of Convexity and Helly-Type Results
Basics
L. Narici, E. Beckenstein, The Hahn-Banach Theorem: The
Life and Times, 2002.
lAÛþw§ù‡½nqLyk‘˜m¥
à8©l5Ÿ3Á‘˜mí2"... |^Ý
˜m5Ÿ5ïĘm5Ÿ´¼©Û¥
˜‡­‡gŽ"– ›Ü§lk‘˜mÃ
‘˜m§lŒÆêÆr•y“êÆ£MmÌ
?¤§pp. 30 – 67, ‰Æч, 2007"
A Glimpse of Convexity and Helly-Type Results
Basics
Half-space Case
Lemma 2
Helly’s Theorem holds when all Ci , i ∈ [n] are closed half
spaces of Rd .
To be continued.
Prove by induction on d. The case of d = 1 is trivial. Assume
d > 1.
A Glimpse of Convexity and Helly-Type Results
Basics
Half-space Case
Lemma 2
Helly’s Theorem holds when all Ci , i ∈ [n] are closed half
spaces of Rd .
To be continued.
Prove by induction on d. The case of d = 1 is trivial. Assume
d > 1. Then induct on n. The case of n ≤ d + 1 is again trivial.
A Glimpse of Convexity and Helly-Type Results
Basics
Half-space Case
Lemma 2
Helly’s Theorem holds when all Ci , i ∈ [n] are closed half
spaces of Rd .
To be continued.
Prove by induction on d. The case of d = 1 is trivial. Assume
d > 1. Then induct on n. The case of n ≤ d + 1 is again trivial.
Thus consider the case of n > d + 1 and assume that
∩i∈[n] Ci = ∅.
(1)
We need to prove that there are ≤ d + 1 Ci which have no
common points, upon the assumption that the lemma is true for
smaller parameters.
A Glimpse of Convexity and Helly-Type Results
Basics
Half-space Case
Lemma 2
Helly’s Theorem holds when all Ci , i ∈ [n] are closed half
spaces of Rd .
To be continued.
Prove by induction on d. The case of d = 1 is trivial. Assume
d > 1. Then induct on n. The case of n ≤ d + 1 is again trivial.
Thus consider the case of n > d + 1 and assume that
∩i∈[n] Ci = ∅.
(1)
We need to prove that there are ≤ d + 1 Ci which have no
common points, upon the assumption that the lemma is true for
smaller parameters.For the smaller parameter case, we could surely
permit some convex sets to be the full space or the empty set.
A Glimpse of Convexity and Helly-Type Results
Basics
finished.
Let π be the hyperplane which borders the half-space Cn and
C 0 the complement of Cn in Rd .
Restricting to the (d − 1)-dimension space π and utilizing the
induction assumption and our ending remark in last slide, we
could assume that
π ∩ (∩i∈[d] Ci ) = ∅.
(2)
If the result is wrong for d and n, then for any i ∈ [n] it occurs
∩j∈[n]\{i} Cj 6= ∅ by virtue of the induction hypothesis. Eq. (1)
tells us that C 0 ∩ (∩i∈[n−1] Ci ) = ∩i∈[n−1] Ci and hence
C 0 ∩ (∩i∈[d] Ci ) 6= ∅.
(3)
By the convexity of ∩i∈[d] Ci , we conclude from Eqs. (2) and
(3) that C ∩ (∩
C ) = ∅. A contradiction.
A Glimpse of Convexity and Helly-Type Results
Basics
We follow “M. Rabin, A note on Helly’s Theorem, Pacific J.
Math. 5 (1955) 363 – 366” to complete a geometric proof of
Helly’s Theorem. For this, we need
Theorem 3 (Finite Basis Theorem, Minkowski 1896, Steinitz
1916, Weyl 1935)
P is a polytope (bounded polyhedron, bounded intersection of
a finite number of closed half-spaces) if and only if it is the
convex hull of a finite number of points (its vertices).
A Glimpse of Convexity and Helly-Type Results
Basics
>.þz˜:“pu±Œ”“ÛÜà5”†
‡>.“o±Ñ”“Nà5”´˜—"Ù
y²ƒØ{ü"à5¯õA^TÐÑ´
ddÚå"¨¨¤ä¥§à5§H˜Ñ‡
§1998§1ʐ"
This classical result is an outstanding example of
a fact which is completely obvious to geometric
intuition, but which wields important algebraic
content and is not trivial to prove. – R.T. Rockafellar
A Glimpse of Convexity and Helly-Type Results
Basics
Linear to Nonlinear
We now prove Helly’s Theorem by linearization.
Lemma 2 + Theorem 3 ⇒ Theorem 1.
For each (d + 1)-subset S of [n], choose a point
vS ∈ ∩i∈S Ci . For each Ci , put
Bi = Conv ({vS : i ∈ S, |S| = d + 1}). (
Ci
Bi .) It is enough to show that
∩i∈[n] Bi 6= ∅.
à§k‚5>
ÏL?n A Glimpse of Convexity and Helly-Type Results
Basics
Linear to Nonlinear
We now prove Helly’s Theorem by linearization.
Lemma 2 + Theorem 3 ⇒ Theorem 1.
For each (d + 1)-subset S of [n], choose a point
vS ∈ ∩i∈S Ci . For each Ci , put
Bi = Conv ({vS : i ∈ S, |S| = d + 1}). (
Ci
Bi .) It is enough to show that
∩i∈[n] Bi 6= ∅.
Verify the above by Lemma 2 and Theorem 3.
à§k‚5>
ÏL?n A Glimpse of Convexity and Helly-Type Results
Basics
Carathéodory’s Theorem
Theorem 4 (Carathéodory’s Theorem)
Any convex combination Q of a set of points
A = {P1 , P2 , . . . , Pn } in Rd is a convex combination of at
most d + 1 points in A.
Proof.
Eliminate the variables one by one algebraically.
Geometric intuition: Induction on dimension; the point lies in
the segment connecting an extreme point and a point in a
lower dimension face.
A Glimpse of Convexity and Helly-Type Results
Basics
Duality between Helly and Carathéodory
>.þz˜:“pu±Œ”“ÛÜà5”†‡>
. o±Ñ”“Nà5”´˜—"
Recall:
“
A Glimpse of Convexity and Helly-Type Results
Basics
Duality between Helly and Carathéodory
>.þz˜: pu±Œ ÛÜà5 †‡>
. o±Ñ Nà5 ´˜—"
Recall:
“
” “
”
“
” “
”
This gives the following geometric observation:
b Q ∈ Conv (P1 , . . . , Pn ) iff ∩i∈[n] Ci = ∅ where Ci is the
closed half-space {x ∈ Rd : (x − Q) · (Pi − Q) ≥ |Pi − Q|2 }.
A Glimpse of Convexity and Helly-Type Results
Basics
Duality between Helly and Carathéodory
>.þz˜: pu±Œ ÛÜà5 †‡>
. o±Ñ Nà5 ´˜—"
Recall:
“
” “
”
“
” “
”
This gives the following geometric observation:
b Q ∈ Conv (P1 , . . . , Pn ) iff ∩i∈[n] Ci = ∅ where Ci is the
closed half-space {x ∈ Rd : (x − Q) · (Pi − Q) ≥ |Pi − Q|2 }.
Lemma 2 (Helly) ⇒ Theorem 4 (Carathéodory).
We still follow “ M. Rabin, A note on Helly’s Theorem, Pacific J.
Math. 5 (1955) 363 – 366”.
By b, ∩i∈[n] Ci = ∅. Consequently, Lemma 2 proves the result.
A Glimpse of Convexity and Helly-Type Results
Basics
Constantin Carathéodory (Sept. 13, 1873 – Feb. 2, 1950)
Carathéodory made significant contributions to the calculus of
variations, the theory of point set measure, and the theory of
functions of a real variable. Carathéodory’s Theorem arises
from his work on power series and harmonic analysis.
A Glimpse of Convexity and Helly-Type Results
Basics
Duality in Hypergraph
Helly hypergraph and conformal hypergraph
A hypergraph H is Helly ⇔ The dual H ∗ of H is conformal
Hypergraph H is acyclic ⇔ H ∗ is Helly and the line graph of
H ∗ is chordal.
A Glimpse of Convexity and Helly-Type Results
Basics
Duality in Hypergraph
Helly hypergraph and conformal hypergraph
A hypergraph H is Helly ⇔ The dual H ∗ of H is conformal
Hypergraph H is acyclic ⇔ H ∗ is Helly and the line graph of
H ∗ is chordal.
C. Beeri, R. Fagin, D. Maier, M. Yannakakis, On the
desirability of acyclic database schemes, J. Assoc. Comput.
Mach., 30 (1983), 479–513.
T.T. Lee, T.Y. Lo, J. Wang, An information-lossless
decomposition theory of relational information systems, IEEE
Trans. on Information Theory, 52 (2006), 1890–1903.
A Glimpse of Convexity and Helly-Type Results
Basics
The universe is built on a plan of profound
symmetry of which is somehow present in the inner
structure of our intellect. – Paul Valery (1871-1945),
French poet and thinker
A Glimpse of Convexity and Helly-Type Results
Basics
Hilbert’s 17th Problem
Let H2k,n be the real space of all homogeneous polynomials of
degree 2k in n given variables.
Theorem 5
Let p ∈ H2k,n be a positive polynomial. Then there exist a
positive integer P
s and vectors c1 , . . . , cm ∈ Rn such that
2s
|x|2s−2k p(x) = m
i=1 (ci · x) .
Proof.
Use Carathéodory’s theorem. See: B. Reznick, Uniform
denominators in Hilbert’s seventeenth problem, Math. Z. 220
(1995), 75–97.
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 6 (Radon’s Lemma)
For any p1 , p2 , . . . , pd+2 ∈ Rd , we can partition [d + 2] into
two disjoint sets A and B such that
Conv ({pi : i ∈ A}) ∩ Conv {pi : i ∈ B} =
6 ∅.
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 6 (Radon’s Lemma)
For any p1 , p2 , . . . , pd+2 ∈ Rd , we can partition [d + 2] into
two disjoint sets A and B such that
Conv ({pi : i ∈ A}) ∩ Conv {pi : i ∈ B} =
6 ∅.
Algebraic proof.
Consider the following d + 1 homogeneous linear equations in
d + 2 variables γ1 , . . . , γd+2 :
γ1 p1 + . . . + γd+2 pd+2 = 0, γ1 + . . . + γd+2 = 0.
(4)
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 6 (Radon’s Lemma)
For any p1 , p2 , . . . , pd+2 ∈ Rd , we can partition [d + 2] into
two disjoint sets A and B such that
Conv ({pi : i ∈ A}) ∩ Conv {pi : i ∈ B} =
6 ∅.
Algebraic proof.
Consider the following d + 1 homogeneous linear equations in
d + 2 variables γ1 , . . . , γd+2 :
γ1 p1 + . . . + γd+2 pd+2 = 0, γ1 + . . . + γd+2 = 0.
(4)
Clearly, there are real numbers x1 , . . . , xd+2 , which are not all
zeros, such that γi = xi , i ∈ [d + 2] are a set of solution to Eq.
(4).
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 6 (Radon’s Lemma)
For any p1 , p2 , . . . , pd+2 ∈ Rd , we can partition [d + 2] into
two disjoint sets A and B such that
Conv ({pi : i ∈ A}) ∩ Conv {pi : i ∈ B} =
6 ∅.
Algebraic proof.
Consider the following d + 1 homogeneous linear equations in
d + 2 variables γ1 , . . . , γd+2 :
γ1 p1 + . . . + γd+2 pd+2 = 0, γ1 + . . . + γd+2 = 0.
(4)
Clearly, there are real numbers x1 , . . . , xd+2 , which are not all
zeros, such that γi = xi , i ∈ [d + 2] are a set of solution to Eq.
(4). Put A = {i ∈ [d + 2] : xi > 0} and B = [d + 2] \ A.
A Glimpse of Convexity and Helly-Type Results
Basics
Radon’s Lemma and Separoid
A Glimpse of Convexity and Helly-Type Results
Basics
Another Proof of Helly’s Theorem
Proof of Theorem 1.
We proceed by induction on n. The base case of n = d + 1 is
trivial. Assume that the result holds for n − 1 and consider the
case for n > d + 1.
A Glimpse of Convexity and Helly-Type Results
Basics
Another Proof of Helly’s Theorem
Proof of Theorem 1.
We proceed by induction on n. The base case of n = d + 1 is
trivial. Assume that the result holds for n − 1 and consider the
case for n > d + 1. For each i ∈ [n], the induction assumption
allows us to take a point pi ∈ ∩j6=i Cj . We are done if there are
i 6= j with pi = pj .
A Glimpse of Convexity and Helly-Type Results
Basics
Another Proof of Helly’s Theorem
Proof of Theorem 1.
We proceed by induction on n. The base case of n = d + 1 is
trivial. Assume that the result holds for n − 1 and consider the
case for n > d + 1. For each i ∈ [n], the induction assumption
allows us to take a point pi ∈ ∩j6=i Cj . We are done if there are
i 6= j with pi = pj . Otherwise, as n ≥ d + 2, Radon’s Lemma
shows that w.l.o.g., there is 1 ≤ t ≤ n − 1 and x ∈ Rd such
that x ∈ Conv ({p1 , . . . , pt }) ∩ Conv ({pt+1 , . . . , pn }).
§
A Glimpse of Convexity and Helly-Type Results
Basics
Another Proof of Helly’s Theorem
Proof of Theorem 1.
We proceed by induction on n. The base case of n = d + 1 is
trivial. Assume that the result holds for n − 1 and consider the
case for n > d + 1. For each i ∈ [n], the induction assumption
allows us to take a point pi ∈ ∩j6=i Cj . We are done if there are
i 6= j with pi = pj . Otherwise, as n ≥ d + 2, Radon’s Lemma
shows that w.l.o.g., there is 1 ≤ t ≤ n − 1 and x ∈ Rd such
that x ∈ Conv ({p1 , . . . , pt }) ∩ Conv ({pt+1 , . . . , pn }).
4 x ∈ Conv ({p1 , . . . , pt }) ⇒ x ∈ ∩i>t Ci ;
§
A Glimpse of Convexity and Helly-Type Results
Basics
Another Proof of Helly’s Theorem
Proof of Theorem 1.
We proceed by induction on n. The base case of n = d + 1 is
trivial. Assume that the result holds for n − 1 and consider the
case for n > d + 1. For each i ∈ [n], the induction assumption
allows us to take a point pi ∈ ∩j6=i Cj . We are done if there are
i 6= j with pi = pj . Otherwise, as n ≥ d + 2, Radon’s Lemma
shows that w.l.o.g., there is 1 ≤ t ≤ n − 1 and x ∈ Rd such
that x ∈ Conv ({p1 , . . . , pt }) ∩ Conv ({pt+1 , . . . , pn }).
4 x ∈ Conv ({p1 , . . . , pt }) ⇒ x ∈ ∩i>t Ci ;
4 x ∈ Conv ({pt+1 , . . . , pn }) ⇒ x ∈ ∩i≤t Ci .
§
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 7 (Kirchberger)
Suppose that there is a finite set R of red points and a finite
set B of blue points in Rd . Suppose that for any set S ⊆ Rd
of d + 2 points there exists a hyperplane which strictly
separates S ∩ R and S ∩ B. Then there exists a hyperplane
which strictly separates R and B.
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 7 (Kirchberger)
Suppose that there is a finite set R of red points and a finite
set B of blue points in Rd . Suppose that for any set S ⊆ Rd
of d + 2 points there exists a hyperplane which strictly
separates S ∩ R and S ∩ B. Then there exists a hyperplane
which strictly separates R and B.
Proof.
For each r ∈ R, consider
Ar = {(c, α) : c ∈ Rd , α ∈ R, c · r < α} ⊆ Rd+1 and for each
b ∈ B put Ab = {(c, α) : c ∈ Rd , α ∈ R, c · b > α} ⊆ Rd+1 .
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 7 (Kirchberger)
Suppose that there is a finite set R of red points and a finite
set B of blue points in Rd . Suppose that for any set S ⊆ Rd
of d + 2 points there exists a hyperplane which strictly
separates S ∩ R and S ∩ B. Then there exists a hyperplane
which strictly separates R and B.
Proof.
For each r ∈ R, consider
Ar = {(c, α) : c ∈ Rd , α ∈ R, c · r < α} ⊆ Rd+1 and for each
b ∈ B put Ab = {(c, α) : c ∈ Rd , α ∈ R, c · b > α} ⊆ Rd+1 .
We need to prove that ∩x∈B∪R Ax 6= ∅, which follows from
Helly’s Theorem.
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 8 (Radius Theorem)
A family of points in Rd is contained in a unit ball if and only
if every subfamily of d + 1 points are contained in a unit ball.
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 8 (Radius Theorem)
A family of points in Rd is contained in a unit ball if and only
if every subfamily of d + 1 points are contained in a unit ball.
Proof.
x1 , x2 , . . . , xn are contained in a unit ball iff
∩ni=1 B(xi , 1) 6= ∅.
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 8 (Radius Theorem)
A family of points in Rd is contained in a unit ball if and only
if every subfamily of d + 1 points are contained in a unit ball.
Proof.
x1 , x2 , . . . , xn are contained in a unit ball iff
∩ni=1 B(xi , 1) 6= ∅.
Make use of Helly Theorem.
A Glimpse of Convexity and Helly-Type Results
Basics
Exercise 9
In a tree, if F1 , . . . , Fk are subtrees and for every i, j, Fi and Fj
share a vertex, then all the Fi share a vertex.
Lehel.
A Glimpse of Convexity and Helly-Type Results
Basics
Exercise 9
In a tree, if F1 , . . . , Fk are subtrees and for every i, j, Fi and Fj
share a vertex, then all the Fi share a vertex.
Lehel.
We prove the contrapositive.
A Glimpse of Convexity and Helly-Type Results
Basics
Exercise 9
In a tree, if F1 , . . . , Fk are subtrees and for every i, j, Fi and Fj
share a vertex, then all the Fi share a vertex.
Lehel.
We prove the contrapositive. If each vertex v misses some tree
Ft(v ) ,
A Glimpse of Convexity and Helly-Type Results
Basics
Exercise 9
In a tree, if F1 , . . . , Fk are subtrees and for every i, j, Fi and Fj
share a vertex, then all the Fi share a vertex.
Lehel.
We prove the contrapositive. If each vertex v misses some tree
Ft(v ) , we mark the edge that leaves v on the unique path to
Ft(v ) .
A Glimpse of Convexity and Helly-Type Results
Basics
Exercise 9
In a tree, if F1 , . . . , Fk are subtrees and for every i, j, Fi and Fj
share a vertex, then all the Fi share a vertex.
Lehel.
We prove the contrapositive. If each vertex v misses some tree
Ft(v ) , we mark the edge that leaves v on the unique path to
Ft(v ) . The tree has one more edge than vertices and so
A Glimpse of Convexity and Helly-Type Results
Basics
Exercise 9
In a tree, if F1 , . . . , Fk are subtrees and for every i, j, Fi and Fj
share a vertex, then all the Fi share a vertex.
Lehel.
We prove the contrapositive. If each vertex v misses some tree
Ft(v ) , we mark the edge that leaves v on the unique path to
Ft(v ) . The tree has one more edge than vertices and so some
edge uw must be marked twice.
A Glimpse of Convexity and Helly-Type Results
Basics
Exercise 9
In a tree, if F1 , . . . , Fk are subtrees and for every i, j, Fi and Fj
share a vertex, then all the Fi share a vertex.
Lehel.
We prove the contrapositive. If each vertex v misses some tree
Ft(v ) , we mark the edge that leaves v on the unique path to
Ft(v ) . The tree has one more edge than vertices and so some
edge uw must be marked twice. This shows that Ft(v ) and
Ft(w ) have no common vertex.
A Glimpse of Convexity and Helly-Type Results
Basics
Exercise 9
In a tree, if F1 , . . . , Fk are subtrees and for every i, j, Fi and Fj
share a vertex, then all the Fi share a vertex.
Lehel.
We prove the contrapositive. If each vertex v misses some tree
Ft(v ) , we mark the edge that leaves v on the unique path to
Ft(v ) . The tree has one more edge than vertices and so some
edge uw must be marked twice. This shows that Ft(v ) and
Ft(w ) have no common vertex.
Exercise 10 (Infinite version of Helly’s Theorem)
Let S be a family of compact convex sets in Rd . If every d + 1
of them have a nonempty intersection, then ∩C ∈S C 6= ∅.
A Glimpse of Convexity and Helly-Type Results
Basics
Exercise 11
Consider n ≥ 4 parallel line segments in R2 . If every three of
these line segments meet a line, then all these line segments
meet a line.
Exercise 12 (Chebyshev approximation)
Let T be a finite set, > 0, and g , fi , i ∈ [m] be m real
functions on T . Suppose that for any (m + 1)-subset S of T ,
we could construct a linear combination fS of fi , i ∈ [m], such
that |fS (x) − g (x)| < for x ∈ S. Then, there exists a
function f , which is a linear combination of fi , i ∈ [m], such
that |f (x) − g (x)| < for x ∈ T .
A Glimpse of Convexity and Helly-Type Results
Basics
The truth always turns out to be simpler than
you thought. – Richard Feynman
Nash-Williams and others popularized a
mnemonic for such theorems: TONCAS, meaning
“The Obvious Necessary Conditions are Also
Sufficient” – D.B. West, Introduction to Graph
Theory, p. 28, China Machine Press, 2004.
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 13 (Krasnosselsky’s Theorem)
Let K be a compact set in R d . Suppose that for every d + 1
points in K , there’s a point of K from which all these points
are visible in K . Then there’s a point of K from which all of K
is visible.
"Vy– ŠW?
"VA§Òìe§
Uq–«§<Xo"
Uññ§^^§
ºNú$„Ú"
A Glimpse of Convexity and Helly-Type Results
Basics
Theorem 14 (Tverberg’s Theorem)
Every (d + 1)(r − 1) + 1 points in Rd can be partitioned into
r parts such that the convex hulls of these parts have
nonempty intersection.
There is no known polynomial algorithm to get the Tverberg’s
partition for general r yet. When r = 2, this is trivial
according to the proof of Theorem 6.
D.G. Larman, On sets projectively equivalent to the vertices of
a convex polytope, Bull. London Math. Soc. 4 (1972), 6–12.
A Glimpse of Convexity and Helly-Type Results
Basics
Colorful Results
Theorem 15 (Colored Carathéodory’s Theorem, Bárány)
Let S1 , . . . , Sd+1 be subsets of Rd . If u ∈ ∩i∈[d+1] Conv (Si ),
then there is vi ∈ Si for each i ∈ [d + 1] such that
u ∈ Conv (v1 , . . . , vd+1 ).
Theorem 16 (Colored Helly Theorem, Lovász)
Let A1 , . . . , Ad+1 be nonempty finite families of convex sets in
Rd . Suppose that each choice Ai ∈ Ai , i ∈ [d + 1], we have
∩i∈[d+1] Ai 6= ∅. Then there is i ∈ [d + 1] such that
∩A∈Ai A 6= ∅.
A Glimpse of Convexity and Helly-Type Results
Basics
Imre Barany, Shmuel Onn, Colourful Linear Programming and
Its Relatives, Mathematics of Operations Research, Vol. 22,
No. 3 (Aug., 1997), pp. 550-567
A Glimpse of Convexity and Helly-Type Results
Basics
Courses
Csaba D. Tóth, Topics in Applied Mathematics,
http://www-math.mit.edu/~toth/18325.html
Benny Sudakov, Algebraic Methods in Combinatorics,
http://www.math.princeton.edu/~bsudakov/
algebraic.html
Otfried Cheong, Topics in Computation Theory, http:
//tclab.kaist.ac.kr/~otfried/cs700sp2005/
Jean H. Gallier, Advanced Geometric Methods in
Computer Science,
http://www.cis.upenn.edu/~cis610/home03.html.
Convex Analysis,
http://math.haifa.ac.il/mathsec/syllabi.html
A Glimpse of Convexity and Helly-Type Results
Basics
Courses
Gil Kalai, (Fall 2005, Yale University), Topic in Discrete
Mathematics: Convexity and Linear Programming,
http://www.ma.huji.ac.il/%7Ekalai/
Philip Pennance, Convex Polytopes I
http://pennance.us/home/courses/dm.php
Linear Programming , http:
//www.cs.elte.hu/opres/courses/topics.html
Combinatorics in Concert: for Teaching, Research,
Outreach and Recreation,
http://pcmi.ias.edu/2004/ufp2003.htm
Igor Pak, Topics in Combinatorics: Convex Polytopes and
Combinatorial Geometry,
http://www-math.mit.edu/~pak/courses/318.htm
A Glimpse of Convexity and Helly-Type Results
Basics
Courses
R. Radoicic, Convex and Discrete Geometry, http:
//www.math.rutgers.edu/~rados/index587.html
Jeong-Hyun Kang, Hemanshu Kaul, Convex and Discrete
Geometry, http://www.math.iit.edu/~kaul/
TeachingUIUC/GeometryCourseProposal.html
Zoltan Furedi, Polytopes and Lattice Points,
http://www.math.uiuc.edu/~z-furedi/TAN/
595math_2008.pdf
S. Govindarajan, N. Mustafa, Combinatorial Geometry,
http://www.mpi-inf.mpg.de/~sgovinda/Course/.
Vladlen Koltun, Advanced Geometric Algorithms,
http://vw.stanford.edu/~vladlen/teaching/
2006-spr-cs369a/index.html
A Glimpse of Convexity and Helly-Type Results
Basics
Courses
V.A. Timorin, Convex Sets,
http://www.mccme.ru/ium/f99/convexity.html
C. Caramanis, Convex Optimization: Theory and
Applications, http:
//users.ece.utexas.edu/~cmcaram/EE381V.html
H. Edelsbrunner, Computational Topology, http:
//www.cs.duke.edu/courses/fall06/cps296.1/
Francis Su, Geometric Combinatorics and Polytopes,
http://www.math.hmc.edu/~su/math189/04s/
D.G. Larman, Discrete Geometry and X-ray Tomography,
http://www.ucl.ac.uk/Mathematics/Courses/
0708/3702.pdf
Thomas Hull, Combinatorial Geometry, http://www.
merrimack.edu/~thull/combgeom/combgeom.html
A Glimpse of Convexity and Helly-Type Results
Basics
Books
Jiri Matousek, Bernd Gartner, Understanding and Using
Linear Programming, Springer, 2007.
A. Schrijver, Combinatorial Optimization: Polyhedra and
Efficiency, Springer, 2003.
Kazuo Murota, Discrete Convex Analysis, SIAM, 2003,
Alexander Barvinok, A Course in Convexity, AMS, 2002.
J. Matousek, Lectures on Discrete Geometry, Springer, 2002.
Jean Gallier, Geometric Methods and Applications For
Computer Science and Engineering, Springer, 2000.
¤ä¥§à5§r•êÆmÖ§H˜Ñ‡§1998"
¤ä¥§à©Û§þ°‰ÆEâч§1990"
Peter D. Lax, Linear Algebra, Wiley, 1997.
A Glimpse of Convexity and Helly-Type Results
Basics
Books
Daniel A. Klain, Gian-Carlo Rota, Introduction to Geometric
Probability, Cambridge University Press, 1997.
J. Pach, P.K. Agarwal, Combinatorial Geometry, John Wiley,
New York, 1995.
Roger Webster, Convexity, Oxford University Press, New
York, 1994.
Ketan Mulmuley, Computational Geometry: An Introduction
through Randomized Algorithms, Prentice Hall, Englewood
Cliffs, NJ, 1994.
P.M. Gruber, J.M. Wills, (Eds.) Handbook of Convex
Geometry, Amsterdam, North-Holland, 1993.
M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms
and Combinatorial Optimization, Springer, 1993.
A. Schrijver, Theory of Linear and Integer Programming,
Wiley, 1986.
A Glimpse of Convexity and Helly-Type Results
Basics
The reading of all good books is like a
conversation with the finest people of past centuries.
– René Descartes (1596–1650)
My life may be encapsulated by one of Graham
Greene’s “entertainments” titles: ‘Loser Takes All’.
Since I was thrown out of high school for political
reasons, I was free to study on my own and develop
my own ways of thinking. – Israel Gelfand, (Gelfand
Workshop, Rutgers Univ., May 6, 2002)
A Glimpse of Convexity and Helly-Type Results
Variations
1Ü©
A Glimpse of Convexity and Helly-Type Results
Variations
The problem should be of intrinsic interest in
even a very special form, but should admit of
interesting extensions. In my opinion, a good
problem is sufficiently specific so that even the
specific form is of interest to someone, but of course
it s best if a specific solution inspires further
questions and generalizations. I deal with a specific
case, if a meaningful (i.e., not obvious but not
impossible) one can be found. Then ‘brainstorm,’
looking for natural generalizations and, if possible,
applications. – Victor L. Klee
.
A Glimpse of Convexity and Helly-Type Results
Variations
Intervals
A transversal of a family S of sets is a set which intersects
with every member of S.
Lemma 17 (Perfectness of the comparability graph of an
interval order)
Let S be a set of intervals in R. Denote the maximum number
of pairwise disjoint intervals from S by ω(S) and the minimum
size of a transversal of S by χ(S). Then ω(S) = χ(S).
Proof.
Denote by r (I ) (`(I )) the right (left) endpoint of an interval I .
Let I ∈ S be an interval with maximum right endpoint, say
r (I ) = x. Put S 0 = {J ∈ S : `(J) > x}.
A Glimpse of Convexity and Helly-Type Results
Variations
Intervals
A transversal of a family S of sets is a set which intersects
with every member of S.
Lemma 17 (Perfectness of the comparability graph of an
interval order)
Let S be a set of intervals in R. Denote the maximum number
of pairwise disjoint intervals from S by ω(S) and the minimum
size of a transversal of S by χ(S). Then ω(S) = χ(S).
Proof.
Denote by r (I ) (`(I )) the right (left) endpoint of an interval I .
Let I ∈ S be an interval with maximum right endpoint, say
r (I ) = x. Put S 0 = {J ∈ S : `(J) > x}.We observe that
χ(S 0 ) + 1 = χ(S)
A Glimpse of Convexity and Helly-Type Results
Variations
Intervals
A transversal of a family S of sets is a set which intersects
with every member of S.
Lemma 17 (Perfectness of the comparability graph of an
interval order)
Let S be a set of intervals in R. Denote the maximum number
of pairwise disjoint intervals from S by ω(S) and the minimum
size of a transversal of S by χ(S). Then ω(S) = χ(S).
Proof.
Denote by r (I ) (`(I )) the right (left) endpoint of an interval I .
Let I ∈ S be an interval with maximum right endpoint, say
r (I ) = x. Put S 0 = {J ∈ S : `(J) > x}.We observe that
χ(S 0 ) + 1 = χ(S) and ω(S 0 ) + 1 = ω(S).
A Glimpse of Convexity and Helly-Type Results
Variations
Helly Type Results: Local vs Global
Theorem 18
Let S be a set of intervals in R. If every s + 1 intervals from S
have a transversal of no more than s points, then S itself has
a transversal of size at most s.
Proof.
By Lemma 17, this says that we cannot find in S a set of
s + 1 disjoint intervals if and only if we cannot do it in any
(s + 1)-subset of S, which is obvious.
Exercise 19
Work out a generalization of Theorem 18. One possibility is to
consider general poset rather than merely the interval order as
treated in Lemma 17 and Theorem 18.
A Glimpse of Convexity and Helly-Type Results
Variations
Theorem 20 (Gyárfás, Lehel )
There exists a finite number L(t) such that, for every finite
collection F of pairwise intersecting sets consisting of at most
t intervals each, there is a set of L(t) points meeting each set
in F . L(1) could be chosen to be 1 and L(2) could be chosen
to be 3.
Exercise 21
Read the following: András Gyárfás, Combinatorics of
Intervals. http: // www. math. gatech. edu/ news/
events/ ima/ newag. pdf
A Glimpse of Convexity and Helly-Type Results
Variations
Theorem 22
of A have a
Let A be a family of r -sets. If every r +s
r
transversal of size at most s, then A itself has a transversal of
size no greater than s.
A Glimpse of Convexity and Helly-Type Results
Variations
Theorem 22
of A have a
Let A be a family of r -sets. If every r +s
r
transversal of size at most s, then A itself has a transversal of
size no greater than s.
Proof.
Suppose the theorem does not hold. Then there is a k > r +s
r
such that there is a k-subset B of A which does not have any
transversal of size ≤ s but the minimum size of the transversal
of any (k − 1)-subset of B is ≤ s and hence must be s. For
any r -subset B ∈ B, put f (B) to be an s-subset which is a
transversal of B \ {B}. Then we find that B ∩ f (B) = ∅ and
B ∩ f (B 0 ) 6= ∅ for B 6= B 0 ∈ B. Our Lemma 23 below says
thus that k ≤ r +s
, yielding a contradiction.
r
A Glimpse of Convexity and Helly-Type Results
Variations
Theorem 22
of A have a
Let A be a family of r -sets. If every r +s
r
transversal of size at most s, then A itself has a transversal of
size no greater than s.
Proof.
Suppose the theorem does not hold. Then there is a k > r +s
r
such that there is a k-subset B of A which does not have any
transversal of size ≤ s but the minimum size of the transversal
of any (k − 1)-subset of B is ≤ s and hence must be s. For
any r -subset B ∈ B, put f (B) to be an s-subset which is a
transversal of B \ {B}. Then we find that B ∩ f (B) = ∅ and
B ∩ f (B 0 ) 6= ∅ for B 6= B 0 ∈ B. Our Lemma 23 below says
thus that k ≤ r +s
, yielding a contradiction.
r
A Glimpse of Convexity and Helly-Type Results
Variations
Taking A =
− 1.
be r +s
r
[r +s]
r
, we see that
r +s
r
can not be weakened to
A Glimpse of Convexity and Helly-Type Results
Variations
Taking A =
− 1.
be r +s
r
[r +s]
r
, we see that
r +s
r
can not be weakened to
Lemma 23 (Bollobás)
Let {Ai : i ∈ [m]} and {Bi : i ∈ [m]} be families of r -sets
and s-sets, resp.,
such that Ai ∩ Bj = ∅ if and only if i = j.
r +s
Then m ≤ r
A Glimpse of Convexity and Helly-Type Results
Variations
Taking A =
− 1.
be r +s
r
[r +s]
r
, we see that
r +s
r
can not be weakened to
Lemma 23 (Bollobás)
Let {Ai : i ∈ [m]} and {Bi : i ∈ [m]} be families of r -sets
and s-sets, resp.,
such that Ai ∩ Bj = ∅ if and only if i = j.
r +s
Then m ≤ r
Proof.
Let X = ∪i∈[m] (Ai ∪ Bi ) and choose vectors vx ∈ Rr +s , x ∈ X ,
which are in general positions.
A Glimpse of Convexity and Helly-Type Results
Variations
Taking A =
− 1.
be r +s
r
[r +s]
r
, we see that
r +s
r
can not be weakened to
Lemma 23 (Bollobás)
Let {Ai : i ∈ [m]} and {Bi : i ∈ [m]} be families of r -sets
and s-sets, resp.,
such that Ai ∩ Bj = ∅ if and only if i = j.
r +s
Then m ≤ r
Proof.
Let X = ∪i∈[m] (Ai ∪ Bi ) and choose vectors vx ∈ Rr +s , x ∈ X ,
which V
are in general positions. For each set S ⊆ X , put
VS = x∈S vx .
A Glimpse of Convexity and Helly-Type Results
Variations
Taking A =
− 1.
be r +s
r
[r +s]
r
, we see that
r +s
r
can not be weakened to
Lemma 23 (Bollobás)
Let {Ai : i ∈ [m]} and {Bi : i ∈ [m]} be families of r -sets
and s-sets, resp.,
such that Ai ∩ Bj = ∅ if and only if i = j.
r +s
Then m ≤ r
Proof.
Let X = ∪i∈[m] (Ai ∪ Bi ) and choose vectors vx ∈ Rr +s , x ∈ X ,
which V
are in general positions. For each set S ⊆ X , put
VS = x∈S vx . Our assumption means that VAi ∧ VBj is
nonzero if and only
V if i = j and hence VAi , i ∈ [m], are linearly
independent in r (Rr +s ).
A Glimpse of Convexity and Helly-Type Results
Variations
Taking A =
− 1.
be r +s
r
[r +s]
r
, we see that
r +s
r
can not be weakened to
Lemma 23 (Bollobás)
Let {Ai : i ∈ [m]} and {Bi : i ∈ [m]} be families of r -sets
and s-sets, resp.,
such that Ai ∩ Bj = ∅ if and only if i = j.
r +s
Then m ≤ r
Proof.
Let X = ∪i∈[m] (Ai ∪ Bi ) and choose vectors vx ∈ Rr +s , x ∈ X ,
which V
are in general positions. For each set S ⊆ X , put
VS = x∈S vx . Our assumption means that VAi ∧ VBj is
nonzero if and only
i = j and hence VAi , i ∈ [m],
linearly
Vr ifr +s
Vr arer +s
independent
in
(R
).
But
the
dimension
of
(R
) is
r +s
r +s
and so m ≤ r follows.
r
A Glimpse of Convexity and Helly-Type Results
Variations
Taking A =
− 1.
be r +s
r
[r +s]
r
, we see that
r +s
r
can not be weakened to
Lemma 23 (Bollobás)
Let {Ai : i ∈ [m]} and {Bi : i ∈ [m]} be families of r -sets
and s-sets, resp.,
such that Ai ∩ Bj = ∅ if and only if i = j.
r +s
Then m ≤ r
Proof.
Let X = ∪i∈[m] (Ai ∪ Bi ) and choose vectors vx ∈ Rr +s , x ∈ X ,
which V
are in general positions. For each set S ⊆ X , put
VS = x∈S vx . Our assumption means that VAi ∧ VBj is
nonzero if and only
i = j and hence VAi , i ∈ [m],
linearly
Vr ifr +s
Vr arer +s
independent
in
(R
).
But
the
dimension
of
(R
) is
r +s
r +s
and so m ≤ r follows.
r
A Glimpse of Convexity and Helly-Type Results
Variations
More proofs:
- Lovász’s proof via Laplace expansion: http:
//people.cs.uchicago.edu/%7Elaci/REU07/a10.pdf
- Blokhuis’s proof via resultant:
http://people.cs.uchicago.edu/~laci/REU07/a11.pdf
- A Combinatorial Proof: Theorem 1.3.1, I. Anderson,
Combinatorics of Finite Sets, Dover, 2002.
A Glimpse of Convexity and Helly-Type Results
Variations
More proofs:
- Lovász’s proof via Laplace expansion: http:
//people.cs.uchicago.edu/%7Elaci/REU07/a10.pdf
- Blokhuis’s proof via resultant:
http://people.cs.uchicago.edu/~laci/REU07/a11.pdf
- A Combinatorial Proof: Theorem 1.3.1, I. Anderson,
Combinatorics of Finite Sets, Dover, 2002.
$ What we do in last slide is merely to translate the
elementary matrix proof of Lovász into the Grassmann algebra
language. Besides the use of Grassman algebra, the key to this
proof is a standard technique in algebraic combinatorics, the
so-called dimension argument.
A Glimpse of Convexity and Helly-Type Results
Variations
Exercise 24
Tackle the puzzle problems in: http: // people. cs.
uchicago. edu/ ~ laci/ REU07/ appuzzles. pdf .
Exercise 25
The duality result, Lemma 17, is used in the proof of Theorem
18. What is the underlying duality result utilized in the proof
of Theorem 22?
Exercise 26
Prove the Grassmann-Plück Identity. Find out its relationship
with matroid theory.
A Glimpse of Convexity and Helly-Type Results
Variations
You already met with exterior algebra (Grassmann algebra) in
your course on multivariable calculus and possibly in your linear
algebra course. You will further meet them in your differential
geometry course and the course on representation of algebras.
All mathematicians stand, as Newton said he did,
on the shoulders of giants, but few have come closer
than Hermann Grassmann to creating,
single-handedly, a new subject. ... In Grassmann’s
geometry a product of line segments is again a
higher-dimensional object. It is a return to Euclid,
but to Euclid with a difference, the difference that
had been dreamed of by Leibniz. – D.
Fearnley-Sander, Hermann Grassmann and the
Creation of Linear Algebra, The American
Mathematical Monthly, 86 (1979) 809–817.
A Glimpse of Convexity and Helly-Type Results
Variations
Hermann Günter Grassmann (1809 – 1877)
A Glimpse of Convexity and Helly-Type Results
Variations
Julius Plücker (1801 – 1868)
Plücker co-ordinates was introduced by Julius Plücker in 1844.
Felix Klein was a student of Julius Plücker.
A Glimpse of Convexity and Helly-Type Results
Variations
Theorem 27 (Sylvester-Gallai)
Let X be a finite set of points in the plane. If for any two
points x1 , x2 ∈ X there is a third point x3 ∈ X such that
x1 , x2 , x3 are colinear, then all points of X are colinear.
Exercise 28
Could you figure out any possible high-dimensional
generalization of Theorem 27?
A Glimpse of Convexity and Helly-Type Results
Variations
There are two ways to do great mathematics.
The first is to be smarter than everybody else. The
second way is to be stupider than everybody else –
but persistent. – Raoul Bott
A Glimpse of Convexity and Helly-Type Results
Variations
De Santis (1957): If we are given n ≥ d + 1 − k convex sets in
Rd such that any d + 1 − k of them contain a common
k-dimensional flat, then they all do.
A Glimpse of Convexity and Helly-Type Results
Variations
Theorem 29 (Konrad J. Swanepoel, Proc. Amer. Math. Soc.
127 (1999), 2155-2162)
A hollow axis-aligned box is the boundary of the cartesian
product of compact intervals in Rd . For d ≥ 3, if any 2d of a
collection of hollow axis-aligned boxes have non-empty
intersection, then the whole collection has non-empty
intersection; and if any 5 of a collection of hollow axis-aligned
rectangles in R2 have non-empty intersection, then the whole
collection has non-empty intersection. The values 2d for d ≥ 3
and 5 for d = 2 are the best possible in general.
M. Deza, P. Frankl, A Helly type theorem for hypersurfaces, J.
Comb. Theory A 45 (1987), 27–30.
A Glimpse of Convexity and Helly-Type Results
Variations
A subset A of the vertex set of a graph G is geodesically
convex if all shortest paths of G joining two vertices of A
belong to A.
H.-J. Bandelt, E. Pesch, A Radon theorem for Helly graphs,
Arch. Math., 52 (1989), 95–98.
Graphs as convexity spaces:
http://www-ma3.upc.es/users/pelayo/research/
Convexity/OPORTO_slides.pdf
A Glimpse of Convexity and Helly-Type Results
Variations
Theorem 30 (Morris’ Theorem)
Let Idm be a finite family of sets in Rd such that each member
of Idm is the disjoint union of at most m closed convex sets and
that the intersection of any two members of Idm is still a
member of Idm . Then, ∅ ∈
/ Idm if and only if the intersection of
any m(d + 1) members of Idm is not the empty set.
H.C. Morris, Two pigeon hole principles and unions of
convexly disjoint sets, PhD Thesis, California Institute of
Technology, Pasadena, California, 1973.
N. Amenta, A short proof of an interesting Helly-type theorem,
Discrete & Computational Geometry 15 (1996), 423 – 427.
A Glimpse of Convexity and Helly-Type Results
Variations
Topology (Global Method!)
A cell in Rd is a set which is homeomorphic to a d-ball.
Theorem 31 (Topological Helly Theorem)
Let K be a finite family of closed sets in Rd such that the
intersection of every k members of K is a cell for k ≤ d and is
nonempty for k = d + 1. Then the intersection of all members
of K is a cell.
L. Danzer, B. Grünbaum, V. Klee, Helly’s Theorem and its
relatives, Proceedings of the Symposium on Pure
Mathematics, Vol. 7, Convexity (1963) pages 101–180, AMS.
I. Barany, J. Matousek, A fractional Helly theorem for convex
lattice sets, Adv. Math., 174 (2003), 227–235.
A Glimpse of Convexity and Helly-Type Results
Variations
I. Barany, M. Katchalski, J. Pach, Helly’s Theorem with volumes,
The American Mathematical Monthly, 91 (1984), 362–365.
G. Kalai, R. Meshulam, Intersections of Leray complexes and
regularity of monomial ideals, J. Combinatorial Theory A 113
(2006), 1586–1592.
G. Kalai, R. Meshulam, A topological colorful Helly Theorem, Adv.
Math. 191 (2005), 305–311.
Raghavan Dhandapani, Jacob E. Goodman, Andreas Holmsen,
Richard Pollack, Convexity in topological affine planes, Discrete &
Computational Geometry, 38 (2007), 243–257.
H.E. Debrunner, Helly type theorems derived from basic singular
homology, The American Mathematical Monthly, 77 (1970),
375–380.
Umed H. Karimova, Dušan Repovš, On the topological Helly
theorem, Topology and its Applications, 153 (2006), 1614–1621.
A Glimpse of Convexity and Helly-Type Results
Variations
[jCS³§,=÷@. – HŠ ù `
5nì„"®µ6
Inspiration is needed in geometry, just as much as
in poetry. – Aleksandr Sergeyevich Pushkin
A Glimpse of Convexity and Helly-Type Results
Variations
<ƒ*ßU/!ìA!ú7!Á~!j˧
k§±Ù¦gƒ§ÃØ3"Å¡±
C§Kiö¯¶x±§K–ö"­ƒÛ
•!4%!š~ƒ*§~3ßx§<ƒ¤¾
–觚k“öØU–"k“°§Ø‘±Ž
§,åØvö½ØU–"k“†å§qØ
‘±ø§–ß_V³¾ÃÔ±ƒƒ§½ØU–
",åv±–èؖ§ß<Œ.§3C
k¥¶¦Æ“§ØU–ö§Œ±Ã¥°§
ÙØU.ƒº– Sœ,5iž!ìP6
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
1nܩ
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
If one understands one’s painting in advance, one
might as well not paint anything. – Salvador Dali,
Spanish Surrealist Painter, (1904 – 1989).
But in the new (math) approach, the important
thing is to understand what you’re doing, rather than
to get the right answer. – Tom Lehrer
A theory has only the alternative of being right or
wrong. A model has a third possibility: it may be
right, but irrelevant. – Manfred Eigen (1927 – )
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Manfred Eigen
http://nobelprize.org/nobel_prizes/chemistry/
laureates/1967/eigen-bio.html
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Fundamental Relation of Linear Equalities
Theorem 32
Let A ∈ Fm×n , b ∈ Fm . Ax = b has a solution x ∈ Fn iff there
is no y ∈ Fm such that y > A = 0, y > b 6= 0.
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Fundamental Relation of Linear Equalities
Theorem 32
Let A ∈ Fm×n , b ∈ Fm . Ax = b has a solution x ∈ Fn iff there
is no y ∈ Fm such that y > A = 0, y > b 6= 0.
Proof.
Ax = b ⇔
Span(A, b) = Span(A)
⇔
(Span(A, b))⊥ = (Span(A))⊥
⇔
(Span(A))⊥ \ (Span(A, b))⊥ = ∅
⇔ there is no y with y > A = 0, y > b 6= 0
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Fundamental Relation of Linear Equalities
Theorem 32
Let A ∈ Fm×n , b ∈ Fm . Ax = b has a solution x ∈ Fn iff there
is no y ∈ Fm such that y > A = 0, y > b 6= 0.
Proof.
Ax = b ⇔
Span(A, b) = Span(A)
⇔
(Span(A, b))⊥ = (Span(A))⊥
⇔
(Span(A))⊥ \ (Span(A, b))⊥ = ∅
⇔ there is no y with y > A = 0, y > b 6= 0
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Theorem 32 is actually the fundamental relation (V ⊥ )⊥ = V
in disguise.
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Lights Out Game
* G : a digraph;
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Lights Out Game
* G : a digraph;
V (G )
* X = F2 : the binary linear space consisting of all maps
from V (G ) to the binary field F2 ;
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Lights Out Game
* G : a digraph;
V (G )
* X = F2 : the binary linear space consisting of all maps
from V (G ) to the binary field F2 ;
* A configuration x of G : x ∈ X ;
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Lights Out Game
* G : a digraph;
V (G )
* X = F2 : the binary linear space consisting of all maps
from V (G ) to the binary field F2 ;
* A configuration x of G : x ∈ X ;
* v is on in x: x(v ) = 1; v is off in x: x(v ) = 0;
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Lights Out Game
* G : a digraph;
V (G )
* X = F2 : the binary linear space consisting of all maps
from V (G ) to the binary field F2 ;
* A configuration x of G : x ∈ X ;
* v is on in x: x(v ) = 1; v is off in x: x(v ) = 0;
* Allowable moves: Switch the states of a given configuration
locally, namely choose a vertex v and switch the states of v
and all its neighbors.
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Theorem 33 (Sutner)
Let G be a graph. For the lights out game on G , we can
always transform the all-off configuration into the all-on
configuration. :-)
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Theorem 33 (Sutner)
Let G be a graph. For the lights out game on G , we can
always transform the all-off configuration into the all-on
configuration. :-)
Proof.
Let B be the adjacency matrix of G and put A = B + I . We
intend to prove that there is x such that Ax = 1.
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Theorem 33 (Sutner)
Let G be a graph. For the lights out game on G , we can
always transform the all-off configuration into the all-on
configuration. :-)
Proof.
Let B be the adjacency matrix of G and put A = B + I . We
intend to prove that there is x such that Ax = 1. By Theorem
32, it suffices to show that there is no y ∈ Fm
2 satisfying both
y > A = 0 and y > 1 6= 0.
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Theorem 33 (Sutner)
Let G be a graph. For the lights out game on G , we can
always transform the all-off configuration into the all-on
configuration. :-)
Proof.
Let B be the adjacency matrix of G and put A = B + I . We
intend to prove that there is x such that Ax = 1. By Theorem
32, it suffices to show that there is no y ∈ Fm
2 satisfying both
y > A = 0 and y > 1 6= 0. Suppose such a y exists.
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Theorem 33 (Sutner)
Let G be a graph. For the lights out game on G , we can
always transform the all-off configuration into the all-on
configuration. :-)
Proof.
Let B be the adjacency matrix of G and put A = B + I . We
intend to prove that there is x such that Ax = 1. By Theorem
32, it suffices to show that there is no y ∈ Fm
2 satisfying both
y > A = 0 and y > 1 6= 0. Suppose such a y exists. Then we get
y > (11> − A)y = (y > 1)2 − (y > A)y 6= 0.
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Theorem 33 (Sutner)
Let G be a graph. For the lights out game on G , we can
always transform the all-off configuration into the all-on
configuration. :-)
Proof.
Let B be the adjacency matrix of G and put A = B + I . We
intend to prove that there is x such that Ax = 1. By Theorem
32, it suffices to show that there is no y ∈ Fm
2 satisfying both
y > A = 0 and y > 1 6= 0. Suppose such a y exists. Then we get
2
y > (11> − A)y = (y > 1)P
− (y > A)y 6= 0. However, we also
>
>
have y (11 − A)y = uv ∈E
/ (G ) (y (u)y (v ) + y (v )y (u)) = 0.
u6=v
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Theorem 33 (Sutner)
Let G be a graph. For the lights out game on G , we can
always transform the all-off configuration into the all-on
configuration. :-)
Proof.
Let B be the adjacency matrix of G and put A = B + I . We
intend to prove that there is x such that Ax = 1. By Theorem
32, it suffices to show that there is no y ∈ Fm
2 satisfying both
y > A = 0 and y > 1 6= 0. Suppose such a y exists. Then we get
2
y > (11> − A)y = (y > 1)P
− (y > A)y 6= 0. However, we also
>
>
have y (11 − A)y = uv ∈E
/ (G ) (y (u)y (v ) + y (v )y (u)) = 0.
u6=v
This contradiction establishes the claim.
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Hyperplane Separation Theorem
Hahn-Banach
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
The 2006 John von Neumann Theory Prize is
awarded by the Institute for Operations Research and
the Management Sciences to Martin Grötschel, László
Lovász and Alexander Schrijver for their fundamental
path-breaking work in combinatorial optimization.
Jointly and individually, they have made basic
contributions to the analysis and solution of hard
discrete optimization problems. In particular, their joint
work on geometric algorithms based on the ellipsoid
method of Yudin-Nemirovski and Shor showed the great
power of cutting-plane approaches to such problems and
provided a theoretical justification for the very active
field of polyhedral combinatorics. One of the
fundamental results was the equivalence of separation
and optimization, ...
http: // www. informs. org/ article. php? id= 1246
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Fundamental Theorem of Linear Inequalities
Theorem 34 (Farkas Lemma)
Exactly one of the following two systems has solutions: 1.
Ax = b and x ≥ 0; 2. A> y ≥ 0 and b > y is not nonnegative.
Proof.
Follows directly from the separating hyperplane theorem.
Theorem 35 (Kronecker)
Let A be a full row-rank integer matrix and b an integer
vector. Exactly one of the following two systems has solutions:
1. Ax = b and x is an integer vector 2. A> y is an integer
vector and b > y is not integer.
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
From Farkas and Carathéodory to Helly
Let’s complete a geometric approach to Helly’s Theorem.
Theorem 34 ⇒ Lemma 2.
Theorems 4, 34 ⇒ Theorem 1.
We could assume that each convex set C is a half-space.
For each (d + 1)-subset S of [n], choose a point v ∈ ∩ C .
i
s
i∈S
i
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Farkas’ motivation came not from mathematical economics
nor yet pure mathematics but from physics; as a professor of
Theoretical Physics he was interested in the problem of
mechanical equilibrium and it was these that gave rise to the
need for linear inequalities. In this he was continuing the
classical work of Fourier and Gauss, though Farkas claims to
be the first to appreciate the importance of homogenious
linear inequalities to these problems. – C.G. Broyden, A simple
algebraic proof of Farkas’ Lemma and related theorems,
Optim. Methods Software 8 (1998), 185–199.
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Exercise 36
(W. Blaschke, 1916) The intersection of half-spaces
{x : x > a ≤ d} and {x : x > a ≥ c} such that d ≥ c is called
a slab of thickness d−c
. Prove that if a bounded convex set in
|a|
R n is contained in no slab of thickness less than t, then, for
t−
. (hint:
every positive , this set contains a ball of radius n+1
Chapter 17 of Vas̀ek Chvátal, Linear Programming, W.H.
Freeman and Company, New York, 1983.)
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
ì­YE¦Ã´§7Vs²q˜~"– ºi
5iìÜ~6
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Fundamental Theorem of Polynomial Equalities
Theorem 37 (Hilbert’s Nullstellensatz)
Let P1 , . . . , Pm , R ∈ F(x) be polynomials in d variables
x = (x1 . . . , xd ). Then exactly one of the following holds: 1.
The system of equations P1 (x) = · · · = Pm (x) = 0, R(x) 6= 0
has a solution x ∈ Fd . 2. There exist polynomials
Q1 , . . . , Qm ∈ F[x] and a nonnegative integer r such that
P1 Q1 + · · · + Pm Qm = R r .
N. Alon, Combinatorial Nullstellensatz, Combin. Probab.
Comput. 8 (1999), 7–29.
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Corollary 38
Let P1 , . . . , Pm ∈ F(x) be polynomials in d variables
x = (x1 . . . , xd ). Then exactly one of the following holds: 1.
The system of equations P1 (x) = · · · = Pm (x) = 0 has a
solution x ∈ Fd . 2. There exist polynomials Q1 , . . . , Qm ∈ F[x]
such that P1 Q1 + · · · + Pm Qm = 1.
When you take stuff from one writer, it’s
plagiarism. When you take stuff from many writers,
it’s research. – Wilson Mizner
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
What is the Fundamental Theorem of Polynomial Inequalities?
Is there any generalization of Theorem 5? Helly’s theorem
could be viewed as a consequence of the fundamental theorem
of linear inequalities. Viewing Helly’s theorem as a linear
theorem, what is its polynomial counterpart?
Topic in discrete mathematics: Social Choice Theory.
http://www.ma.huji.ac.il/%7Ekalai/course07.html
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
In the pure mathematics we contemplate
absolute truths which existed in the divine mind
before the morning stars sang together, and which
will continue to exist there when the last of their
radiant host shall have fallen from heaven. – Edward
Everett (1794 – 1865)
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
The Five Platonic Solids (and some friends)
M. C. Escher, Study for Stars, Woodcut, 1948
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
Laplace Expansion and Axiom of Valuated Matroid
A Glimpse of Convexity and Helly-Type Results
Separating hyperplane and theorems of alternatives
The Geometry Junkyard