Pacific Journal of
Mathematics
CONTINUITY AND CONVEXITY OF PROJECTIONS AND
BARYCENTRIC COORDINATES IN CONVEX POLYHEDRA
J OHN A RNOLD K ALMAN
Vol. 11, No. 3
BadMonth 1961
CONTINUITY AND CONVEXITY OF PROJECTIONS AND
BARYCENTRIC COORDINATES IN CONVEX POLYHEDRA
J. A.
KALMAN
If sQ,
, sn are linearly independent points of real ^-dimensional
Euclidean space Rn then each point x of their convex hull S has a (unique)
representation x = Σ?=oMίC)s< with \(%) ^ 0 (i = 0,
, n) and Σ?=<Λ<(#) — 1>
and the barycentric coordinates λ0,
, Xn are continuous convex functions
on S (cf. [3, p. 288]). We shall show in this paper that given any finite
set 809 •• ,s f n of points of Rn we can assign barycentric coordinates
λ>o>
, λTO to their convex hull S in such a way that each coordinate is
continuous on S and that one prescribed coordinate (λ0 say) is convex on
£ (Theorem 2); the author does not know whether it is always possible
to make all the coordinates convex simultaneously (cf. Example 3). In
proving Theorem 2 we shall use certain "projections" which we now
define; these projections are in general distinct from those of [1, p. 614]
n
and [2, p. 12], Given two distinct points s0 and s of R , let sos be the
open half-line consisting of all points sQ + X(s — s0) with λ > 0; given a
n
n
point s0 of R and a closed subset S of R such that s0 $ S, let C(s0, S)
be the "cone" formed by the union of all open half-lines sos with s in
S; and given a point x in such a cone C(sQ, S), let π(x) be the (unique)
point of sox Π S which is closest to s0. Then we shall call the function
π the "projection of C(so,S) on S." Our proof of Theorem 2 depends
on the fact that if S is a convex polyhedron then π is continuous (Theorem 1). This result may appear to be obvious, but it is not immediately
obvious how a formal proof should be given; moreover, as we shall show
(Examples 1 and 2), the conclusion need not remain true for polyhedra
S which are not convex or for convex sets S which are not polyhedra.
The author is indebted to the referee for improvements to Lemma 3,
Example 1, and Example 2, and for the remark at the end of § 1.
1. Projections* For any subset A of Rn we shall denote by H(A)
the convex hull of A and by L(A) the affine subspace of Rn spanned by
A (cf. [2, pp. 21, 15]). If A = {s19
, sp} we shall write H(A) =
n
H(slf
, sp) and L(A) = L(slf
, sp). Given two points x and y of R
we shall denote by (x, y) the inner product of x and y and by \x — y\
the Euclidean distance V(x — y, x — y) between x and y.
n
1. Let s0 he a point of R , let S be a closed convex subset
n
of R such that s0 0 S, and let π be the projection of C(s0, S) on S.
Suppose that points x, slf
, sp of S and real numbers \,
, Xp are
LEMMA
Received July 7, 1960.
1017
1018
J. A. KALMAN
such that x = Σf = 1 λA, Xt > 0 (i = 1,
, p), ΣiLi\
= 1, α^cί π(a?) = x.
Then
(i)
(ii)
π(y) = 7/ / o r αM # m iffo,
So^Lfo, . . . , 8 , ) .
, s p ); and
Proof. ( i ) Given 7/ in H(slf
, sp) we can find nonnegative real
numbers μu
, μp such that y = Σ<=iiM»βi and Σ £ i / ^ = l
Since each
λ< > 0, there exists α: with 0 < α: < 1 such that λ; — α:/^ > 0 for each
ΐ = 1, . . . , p. Let
2-
x
-
a y
- v ί2^i_^M
then z e i?(Si, •• , 8 ί ) g S and a; = α?/ + (1 — α:)s. We now use an indirect argument. Suppose that π(y) Φ y; then for some β with 0 < β < 1
we have π(y) = (1 — /5)s0 + /5i/ and
/1 \
α:(l - β)80 + βx
α(l - /8) + β
=
aπ(y) + β(l — a)z =
a + β(l~a)
f
say. It follows from (1) that xf e sox n S and that \s0 — xf \ < | s0 — x |,
contradicting the hypothesis that 7r(#) = aj. This completes the proof of
(i).
(ii) Suppose that s0 e L(su
, sp). Then we can find real numbers
^i> •••> ^p such that s0 = Σιί=iViSί and Σf=i^< ~ l
Since each λ^ > 0,
there exists 7 with 0 < γ < 1 such that λ< — 7(λi — P<) > 0 for each
i = 1,
, p. But then if
w = γs 0 + (1 - 7)a = Σ [λ« - 7(λ, - v^s,
i =l
we have w e sox Π *S and | s0 — w \ < \ s0 — x \, contradicting the hypothesis
that π(x) = x. This completes the proof of (ii).
Let s0, S, and π be as in Lemma 1. Then we shall call a subset A
of S "π-admissible" if π(x) = α? for all x in iί(A).
2. Lei s0, S, α^cZ π be as in Lemma 1, let A be a finite
π-admissible subset of S, and let π' be the projection of C(s0, H(A)) on
H(A).
Then
( i ) π(x) = π'(x) for all x in C(s0, H(A)); and
(ii) πf is a continuous mapping of C(s0, H(A)) into H(A).
LEMMA
Proof. ( i ) Let x be any point of C(s0, H(A)). Then π(π'(x)) = π\x)
since A is π -admissible, hence π'(x) is the point of sQπf(x) (Ί S = sQx Π S
which is closest to s0, and hence π(x) = π'(a ).
(ii) Let A = {slf •••, sp} and let a?0 = Σ?=I(1/P) S <; t h e n ^ o ) = χo
CONTINUITY AND CONVEXITY OF PROJECTIONS
1019
since A is π-admissible, and hence s0 φ L(A) by Lemma 1. It follows
that if s* is the point of L(A) which is closest to s0, and x is any point
of C(s0, H(A)), then
π'(x) =
x
~ λ^s°
,
where
X(x) = ^ ~ 8 * ' s ° ~ s*> .
Hence π' is continuous.
3. Let s0 be a point of Rn and let T be a closed bounded
subset of Rn such that s0 0 T. Then {s0} U C(sQ, T) is a closed subset
of R\
With the help of the Bolzano-Weierstrass theorem it is not difficult
to prove Lemma 3.
LEMMA
1. Let s0, su * , s m be points of Rn such that s0 0
H(slf
, sm) — S say, and let π be the projection of C(s0, S) on S. Then
π is a continuous mapping of C(s0, S) into S.
THEOREM
Proof. Let Al9
, Aq be the subsets of {slf
, sm} which are πadmissible subsets of S. Then each x in C(s0, S) belongs to at least one
C(s0, H(Aj)) (1 ^ i g q); indeed, given x in C(s0, S), there exist positive
integers #(1), •• ,αs(p) and positive real numbers Xl9 * , λ p such that
π
(%) = Σf=i λ Λ(ί) and Σ f = i \ = 1>a n d ^ e n A = {sx(1),
, sx(p)} is π-admissible by Lemma 1 (i), and x e C(s0, H(A)). For each j = 1,
, g let TΓ^
be the projection of C{s0, H{A3)) on iίίAj).
To prove the theorem it will be enough to show that, if x, x19 x2,
in C(s0, S) are such that x = limfcxΛ, then it follows that π(x) = limfc^(xfc).
Let / b e the set of alii (1 ^ 3 ^ tf) such that xfc e C(s09 H(Aj)) for infinitely
many values of k, and for each j in J let j ( l ) < j(2) <
be the values
of k such that xk e C(s0, HiAj)).
Now, for each j in J, x e C(s0, H(A5))
by Lemma 3, and hence, by Lemma 2, π(x) = TΓ^X) = lim^^aj^ϊ)) =
lim z π(x j ( z ) ). Since all but a finite number of the positive integers are
of the form j(ΐ) for some j in J and some ί = 1, 2,
, it follows that
π(x) — limfc π(xk), as we wished to prove.
The following example shows that if S is a non-convex polyhedron
in R\ and s0 $ S, then the projection of C(s0, S) on S need not be continuous.
EXAMPLE 1. Let s0 = (0, 2), s, = (0,1), s2 = (0, 0), and s3 = (1, 0);
and let S = ^ ( s ^ s2) U H(s2, s3). Then the projection of C(s0, S) on S is
not continuous at sλ.
The following example shows that if S is a closed convex set in R\
and s0 ^ S, then the projection of C(sQ, S) on S need not be continuous.
1020
J. A. KALMAN
EXAMPLE 2. Let s0 = (0, 0, 2), let sλ = (0, 0,1), let K be the circle
consisting of all points (ξ, η, ζ) in R3 such that (ξ — I) 2 + yf = 1 and
? = 0, let S = fΓ({8j U K\ and let π be the projection of C(s0, S) on S.
Then if we set xk = (1 — cos Ar1, sin ΛΓ1, 0) (k = 1, 2, •) we have xk e
C(s0, S) and π(xΛ) = xk (k = 1, 2, •). When fc -> oo, a?4 -> (0, 0, 0) = s2
say, and τr(a?fc) —> s2; since π(s2) = sx, this shows that 7Γ is not continuous
at 82.
Theorem 1 is valid for each closed convex set S §Ξ R2,
n
and for each strictly convex closed set S g R .
REMARK.
2. Barycentric coordinates. Let s0 be a point of iϋ71, let S be a
closed convex subset of Rn such that s0 0 S, and let Z)(s0, S) be the union
of all segments H(s0, s) joining s0 to points s of S; then D(s 0 , S) is a
convex set. Define a real-valued function λ0 on D(s0, S) as follows: let
λo(so) = 1, let λo(α?) = 0 if x e S, and if x φ sQ and x $ S let λo(α?) be
defined by the equation x = λo(^)so + [1 — λo(α?)]π(a?), where π is the projection of C(80, S) on S; then each x in D(s 0 , S) has a representation of
the form
(2 )
x = XQ(x)s0 + [1 - XQ(x)]s ,
with 8 in S.
We shall call λ0 the "barycentric function of D(s0, S)."
4.
of R such that
Then 0 ^ X0(x)
D(s0, S). If S
LEMMA
n
Let s0 be a point of Rn, let S be a closed convex subset
sQ $ S, and let λ0 be the barycentric function of D(s0, S).
^ 1 for all x in D(s0, S) and λ0 is a convex function on
is a polyhedron then λ0 is continuous on D(sQ, S).
Proof. It is clear that X0(x) ^ 1 for
that X0{x) ^ 0 for all x in D(sOf S) depends
will be left to the reader. To prove that
f
show that if x, x e D(s0, S) and 0 < a < 1
(3)
\(ax
all x in D{s0, S); the proof
on the convexity of S, and
λ0 is convex on D(s0, S) we
then
f
+ (1 - a)x ) ^ aX0(x) + (1 - afyφ') .
f
Let x* = ax + (1 — a)x' and let β = aX0(x) + (1 — a)X0(x ); we may assume
that β < 1 since otherwise (3) is trivial. Then if γ = a[l - λo(a?)](l - β)'\
and s, 8r in S are such that
x = X0(x)sQ + [1 - λo(α?)]s , x' = λo(x')so + [1 - λo(x')]sf
(cf. (2)), we have
(4)
7s + (1 - γ)β' = —/9(1 - /S)-1^ + (1 - βy'x*
and 7s + (1 — y)s' e S since S is convex.
,
It follows from (4) that x* Φ sQ.
CONTINUITY AND CONVEXITY OF PROJECTIONS
1021
If x* $ S and π is the projection of C(s0, S) on S then
π(x*) = -λo(α*)[l - Ux*)]-% + [1 - ^(a?*)]-^* ,
and hence by (4) and the definition of π, \(x*) ^ β, as asserted by (3).
If x* e S then (3) is trivial. This completes the proof that λ0 is convex
on D(s0, S).
We next show that λ0 is continuous at s0. Given ε with 0 < ε < 1,
let δ = Me, where M > 0 is the shortest distance from s0 to S. Then
if a? 6 D(s0, S) and 0 < | x — s0 | < 8 we have x Φ sOf x φ S, and
M g I π(x) — β01 = [1 — λo(x)]-1 \x - s 0 1 < [1 - \ϋ(x)]'1Me ,
and hence 0 < 1 — λo(a;) < ε. This proves that λ0 is continuous at s0.
It remains to prove that λ0 is continuous on JD(S0, S) — {s0} if S is a
polyhedron. For each x in C(s0, S) define //0(a;) by the equation x —
μo(x)so + [1 - μo(x)]π(x); then
(5)
AΦ0 = 1 - l « - β o | / | π ( α ? ) - 8 o | .
It follows that //0(a;) g 0 if x e S, and that μQ(x) = λo(cc) > 0 if cc e Z)(s0, S),
a? =^ 80, and x ψ S; thus
(6)
X0(x) = max [/^oW, 0] (a? e D(s 0 , S), a? ^ s0) .
If S is a polyhedron then μQ is continuous on C(s0, S) by Theorem 1 and
(5), and hence λ0 is continuous on D(sϋy S) — {s0} by (6). This completes
the proof of the lemma.
2. Let s0, •••,«» δβ points of Rn9 and let S = iϊ(s 0 ,
Γfcen ίfeere eίcisί nonnegative real-valued continuous functions λ0,
on S, wife λ0 a convex function, such that, for each x in S,
THEOREM
, s m ).
, λTO
x= 2
ΐ=0
Proof. We use induction on m. The case m = 0 is trivial. We
assume the theorem to have been proved for m = M — 1 and deduce it
for m = M. Let T = H(su
, s^). If s0 e T we may set X0(x) = 0
for all a? in S, and deduce the existence of λx,
, XM directly from the
induction hypothesis; we therefore assume that s0 0 T. By the induction
hypothesis there exist nonnegative real-valued continuous functions
]"i,
,J"jfθnΓ such that, for each y in T, ! / = Σ £ i J"i(w)βt, and ΣS=i μlv) = l
Let λ0 be the barycentric function of D(s 0 , ϊ 1 ) . Then each x in S =
D(s 0 , Γ) has a representation of the form x = λo(^)so + [1 — λo(#)]sx with
sx in Γ (cf. (2)),- and if we now set λ<(a?) = ft(sj[l - λo(a?)] (x e S; i =
1, « ,ikf) then it follows that the λ< (i = 1,
, M) are well-defined
1022
J. A. KALMAN
functions on S, and, by Lemma 4, that the functions λ0,
, XM satisfy
all the conditions in the statement of the theorem.
To show that the functions X{ defined in the proof of Theorem 2
need not all be convex we can let s0, s19 s2, and s3 be the points (0, 2),
(1, 0), (—1, 0), and (0,1) respectively of R2 and let S = H(sQ, su s2, s8);
however in this example we obtain convex barycentric coordinates if we
interchange the roles of s0 and s3. In the following example some of
the barycentric coordinates determined as in the proof of Theorem 2 fail
to be convex no matter how s0 is chosen.
3. Define ί0, •• ,ί 4 in it?3 as follows: ί0 = (0, 0,1), tx =
(0,1,0), U = ( 0 , - 1 , 0 ) , ί8 = ( 1 , 0 , - 1 ) , and t4 = ( - 1 , 0, - 1 ) ; let S =
H(t0, •• ,£ 4 ); and let barycentric coordinates be defined for S as in the
proof of Theorem 2, with
( i ) ίo,
(ii) tλ or ί2, and
(iii) t3 or t4 playing the role of s0. Then if we write θ± for max [ ± # , 0 ]
(θ real) we obtain
EXAMPLE
(i)
(ii)
(iii)
(I, 0, 0) = III to + (J - \ξ\){tι + t2) + ξ+t3 + |_ί 4
(III ^ i) ,
(0, η, 0) = i ( l - I η |)ί0 + 9 Λ + 37-ί, + i ( l - | η |)(t8 + ί4)
( 1 ^ 1 ^ 1), and
(0, 0, ζ) = ζ+to + i ( l - I ? |)(ίi + U) + hξ-(t* + **)
(I f I ^ 1),
respectively, and hence in no case are the barycentric coordinates all
convex.
The argument in the proof of Theorem 2 amounts to determining
barycentric coordinates λ0,
, Xm for H(sQ,
, sm) by first choosing λ 0
as small as possible, then choosing λ t as small as possible with this choice
of λ0, etc. We remark in conclusion that if we first choose λ0 as large
as possible, then choose \ as large as possible with this choice of λ0, etc.,
we do not in general obtain convex barycentric coordinates; this may be
seen by considering the case of a square in R2.
REFERENCES
1. P. Alexandroff and H. Hopf, Topologie I, Berlin, 1935.
2. H. G. Eggleston, Convexity, Cambridge, England, 1958.
3. S. Lefschetz, Algebraic topology, New York, 1942.
UNIVERSITY OF AUCKLAND
PACIFIC JOURNAL OF MATHEMATICS
EDITORS
RALPH S. PHILLIPS
A.
Stanford University
Stanford, California
University of Southern California
Los Angeles 7. California
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BROWNELL
L.
L. J.
University of Washington
Seattle 5, Washington
WHITEMAN
PAIGE
University of California
Los Angeles 24, California
ASSOCIATE EDITORS
E. F. BECKENBACH
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Pacific Journal of Mathematics
Vol. 11, No. 3
BadMonth, 1961
Errett Albert Bishop, A generalization of the Stone-Weierstrass theorem . . . . . . . . . . . 777
Hugh D. Brunk, Best fit to a random variable by a random variable measurable with
respect to a σ -lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785
D. S. Carter, Existence of a class of steady plane gravity flows . . . . . . . . . . . . . . . . . . . . 803
Frank Sydney Cater, On the theory of spatial invariants . . . . . . . . . . . . . . . . . . . . . . . . . . 821
S. Chowla, Marguerite Elizabeth Dunton and Donald John Lewis, Linear
recurrences of order two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833
Paul Civin and Bertram Yood, The second conjugate space of a Banach algebra as
an algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847
William J. Coles, Wirtinger-type integral inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871
Shaul Foguel, Strongly continuous Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879
David James Foulis, Conditions for the modularity of an orthomodular lattice . . . . . . 889
Jerzy Górski, The Sochocki-Plemelj formula for the functions of two complex
variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897
John Walker Gray, Extensions of sheaves of associative algebras by non-trivial
kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909
Maurice Hanan, Oscillation criteria for third-order linear differential equations . . . . 919
Haim Hanani and Marian Reichaw-Reichbach, Some characterizations of a class of
unavoidable compact sets in the game of Banach and Mazur . . . . . . . . . . . . . . . . . 945
John Grover Harvey, III, Complete holomorphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 961
Joseph Hersch, Physical interpretation and strengthing of M. Protter’s method for
vibrating nonhomogeneous membranes; its analogue for Schrödinger’s
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971
James Grady Horne, Jr., Real commutative semigroups on the plane . . . . . . . . . . . . . . . 981
Nai-Chao Hsu, The group of automorphisms of the holomorph of a group . . . . . . . . . . 999
F. Burton Jones, The cyclic connectivity of plane continua . . . . . . . . . . . . . . . . . . . . . . . . 1013
John Arnold Kalman, Continuity and convexity of projections and barycentric
coordinates in convex polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017
Samuel Karlin, Frank Proschan and Richard Eugene Barlow, Moment inequalities of
Pólya frequency functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023
Tilla Weinstein, Imbedding compact Riemann surfaces in 3-space . . . . . . . . . . . . . . . . . 1035
Azriel Lévy and Robert Lawson Vaught, Principles of partial reflection in the set
theories of Zermelo and Ackermann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045
Donald John Lewis, Two classes of Diophantine equations . . . . . . . . . . . . . . . . . . . . . . . 1063
Daniel C. Lewis, Reversible transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
Gerald Otis Losey and Hans Schneider, Group membership in rings and
semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089
M. N. Mikhail and M. Nassif, On the difference and sum of basic sets of
polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099
Alex I. Rosenberg and Daniel Zelinsky, Automorphisms of separable algebras . . . . . 1109
Robert Steinberg, Automorphisms of classical Lie algebras . . . . . . . . . . . . . . . . . . . . . . . 1119
Ju-Kwei Wang, Multipliers of commutative Banach algebras . . . . . . . . . . . . . . . . . . . . . 1131
Neal Zierler, Axioms for non-relativistic quantum mechanics . . . . . . . . . . . . . . . . . . . . . 1151