12
DISCRETE ASPECTS OF STOCHASTIC
GEOMETRY
Rolf Schneider
INTRODUCTION
Stochastic geometry studies randomly generated geometric objects. The present
chapter deals with some discrete aspects of stochastic geometry. We describe work
that has been done on familiar objects of discrete geometry, like finite point sets,
their convex hulls, discrete point sets, arrangements of flats, tessellations of space,
under various assumptions of randomness. Most of the results to be mentioned
concern expectations of geometrically defined random variables, or probabilities of
events defined by random geometric configurations. The selection of topics must
necessarily be restrictive. We leave out the great number of special elementary
geometric probability problems which can be solved explicitly by direct, though
possibly intricate, analytic calculations. We pay special attention to either asymptotic results, where the number of considered points tends to infinity, or inequalities,
or identities where the proofs involve more delicate geometric or combinatorial arguments. The close ties of discrete geometry to convexity are reflected: we consider
convex hulls of random points, intersections of random halfspaces, tessellations of
space into convex sets, generated either by discrete random hyperplane systems or,
as Voronoi or Delaunay mosaics, by discrete random point sets. Topics not covered
are, for example, optimization problems with random data, and the average-case
analysis of geometric algorithms.
12.1 CONVEX HULLS OF RANDOM POINTS
The setup for this section is a finite number of random points in a topological
space Σ. Mostly the space Σ is Rd , the d-dimensional Euclidean space, with scalar
product
h·, ·i and norm k · k. Other spaces that occur are the sphere S d−1
Œ
Œ := {x ∈
Rd Œ kxk = 1} or more general submanifolds of Rd . By B d := {x ∈ Rd Œ kxk ≤ 1}
we denote the unit ball of Rd . The volume of B d is denoted by κd .
GLOSSARY
Random point in Σ:
into Σ.
A Borel measurable mapping from some probability space
Distribution of a random point X in Σ : The probability measure µ on Σ such
that µ(B), for a Borel set B ⊂ Σ, is the probability that X ∈ B.
i.i.d. random points: Stochastically independent random points (on the same
probability space) with the same distribution.
2
R. Schneider
NOTATION
X1 , . . . , Xn
µ
ϕ
ϕ(µ, n)
ϕ(K, n)
E
fk
ψj
Vj
Vd
S
Dj (K, n)
i.i.d. random points in Rd
the common probability distribution of Xi
a measurable real function defined on polytopes in Rd
the random variable ϕ(conv{X1 , . . . , Xn })
= ϕ(µ, n), if µ is the uniform distribution in K
expectation of a random variable
number of k-faces
1 on polytopes with j vertices, 0 otherwise
jth intrinsic volume (see Chapter 16); in particular:
d-dimensional volume
= 2Vd−1 , surface area; dS element of surface area
= Vj (K) − Vj (K, n)
12.1.1 DISTRIBUTION-INDEPENDENT RESULTS
There are a few general results on convex hulls of i.i.d. random points in Rd that
do not require special assumptions on the distribution of these points. A clas/
sical result due to Wendel [Wen62] concerns the probability, say pd,n , that 0 ∈
conv{X1 , . . . , Xn }. If the distribution of the i.i.d. random points X1 , . . . , Xn ∈ Rd
is symmetric with respect to 0 and assigns measure zero to every hyperplane through
0, then
“
d−1 ’
1 X n−1
pd,n = n−1
(12.1.1)
.
2
k
k=0
This follows from a combinatorial result of Schläfli, on the number of d-dimensional
cells in a partition of Rd by n hyperplanes through 0 in general position. It was
proved surprisingly late that the symmetric distributions are extremal: Wagner
and Welzl [WaW01] showed that if the distribution of the points is absolutely
continuous with respect to Lebesgue measure, then pd,n is at least the right-hand
side of (12.1.1).
The expected values EVd (µ, n) for different numbers n are connected by a sequence of identities. For an arbitrary probability distribution µ on Rd , Buchta
[Buc90] proved the recurrence relations
EVd (µ, d + 2m) =
“
’
2m−1
d + 2m
1 X
EVd (µ, d + 2m − k)
(−1)k+1
k
2
k=1
and, consequently,
EVd (µ, d + 2m) =
m
X
€
k=1
2
2k
“
’
 B2k d + 2m
−1
EVd (µ, d + 2m − 2k + 1)
k
2k − 1
for m ∈ N, where the constants B2k are the Bernoulli numbers.
12.1.2 NATURAL DISTRIBUTIONS
In geometric problems about random points, a few distributions have been considered as particularly natural, for different reasons. Such reasons may be invariance
Discrete aspects of stochastic geometry
3
properties, or relations to measures of geometric significance, but there are also
more subtle viewpoints, as explained, for example, in Ruben and Miles [RuM80].
The distributions of a random point in Rd shown in Table 12.1.1 underlie many
investigations.
TABLE 12.1.1 Natural distributions of a random point in Rd .
NAME OF DISTRIBUTION
Uniform in K
Standard normal
Beta type 1
Beta type 2
Spherically symmetric
PROBABILITY DENSITY AT x ∈ R
d
function
of K at x
∝ indicator
€

∝ exp − 21 kxk2
∝ (1 − kxk2 )q × indicator function of B d at x, q > −1
∝ kxkα−1 (1 + kxk)−(α+β) , α, β > 0
function of kxk
Here K ⊂ Rd is a given closed set of positive, finite volume, often a convex
body (a compact, convex set with interior points). Usually the name of the distribution of a random point is also associated with the random point itself. General
rotationally symmetric distributions have mostly been considered under additional
tail assumptions. If F is a smooth compact hypersurface in Rd , a random point
is uniform on F if its distribution is proportional to the area measure on F . This
distribution is particularly natural for the unit sphere S d−1 , since it is the unique
rotation-invariant probability measure on S d−1 .
For combinatorial problems about n-tuples of random points in Rd , the following approach leads to a natural distribution. Every configuration of n > d
numbered points in general position in Rd is affinely equivalent to the orthogonal
projection of the set of numbered vertices of a fixed regular simplex T n−1 ⊂ Rn−1
onto a unique d-dimensional linear subspace of Rn−1 . This establishes a one-toone correspondence between the (orientation-preserving) affine equivalence classes
of such configurations and an open dense subset of the Grassmannian G(n − 1, d)
of oriented d-spaces in Rn−1 . The unique rotation-invariant probability measure
on G(n − 1, d) thus leads to a probability distribution on the set of affine equivalence classes of n-tuples of points in general position in Rd . References for this
Grassmann approach, which was proposed by Vershik and by Goodman and
Pollack, are given in Affentranger and Schneider [AfS92]. Baryshnikov and Vitale
[BaV94] proved that an affine-invariant functional of n-tuples with this distribution
is stochastically equivalent to the same functional taken at an i.i.d. n-tuple of standard normal points in Rd . Baryshnikov [Bar97] has made clear, in a strong sense,
the unique role that is played in this correspondence by the vertex sets of regular
simplices.
12.1.3 UNIFORM RANDOM POINTS IN CONVEX BODIES
A considerable amount of work has been done on convex hulls of a finite number
of i.i.d. random points with uniform distribution in a given convex body K in Rd .
Some of the expectations of ϕ(K, n) for different functions ϕ are connected by
4
R. Schneider
identities. Two classical results of Efron [Efr65],
’
“
EVdn−d−1 (K, d + 1)
n
Eψd+1 (K, n) =
d+1
Vdn−d−1 (K)
and
Ef0 (K, n + 1) =
n+1
EDd (K, n),
Vd (K)
(12.1.2)
(12.1.3)
have found far-reaching generalizations in work of Buchta [Buc02]. He extended
(12.1.3) to higher moments of the volume, showing that
“
k ’
Y
f0 (K, n + k)
EVdk (K, n)
1
=
−
E
n+i
Vdk (K)
i=1
for k ∈ N. As a consequence, the kth moment of Vd (K, n) can be expressed linearly
by the first k moments of f0 (K, n+k). Further consequences are variance estimates
for Dd (K, n) and f0 (K, n) for sufficiently smooth convex bodies K.
Of combinatorial interest is the expectation Eψi (K, n), which is the probability
that the convex hull of n i.i.d. uniform random points in K has exactly i vertices.
For this, Buchta [Buc02] proved that
Eψi (K, n) = (−1)i
’ “
’ “X
i
i EVdn−j (K, j)
n
(−1)j
,
i j=1
j
Vdn−j (K)
which for i = d + 1 reduces to (12.1.2). Sylvester’s classical problem asked for
Eψ3 (K, 4) (or the complementary probability) for a convex body K ⊂ R2 . More
generally, one may ask for Eψd+1 (K, n) for a convex body K ⊂ Rd and n > d + 1,
the probability that the convex hull of n i.i.d. uniform points in K is a simplex.
From (12.1.2) and results of Miles [Mil71b], the values Eψd+1 (B d , n) are known.
At the other end, Eψn (K, n) is of interest, the probability that n i.i.d. uniform
points in K are in convex position. Valtr [Val95] determined Eψn (P, n) if P is a
parallelogram, and in [Val96] if P is a triangle. For convex bodies K ⊂ R2 of area
one, Bárány [Bár99] obtained the astonishing limit relation
lim n2
n→∞
p
n
Eψn (K, n) =
1 2 3
e A (K),
4
where A(K) is the supremum of the affine perimeters of all convex bodies contained
in K. Bárány has even established a law of large numbers for convergence to a limit
shape. There is a unique convex body K̃ ⊂ K with affine perimeter A(K). If Kn
denotes the convex hull of n i.i.d. uniform points in K and δ is the Hausdorff
metric, then Bárány’s result says that
lim Prob{δ(Kn , K̃) >  | f0 (Kn ) = n} = 0
n→∞
for every  > 0.
For balls in increasing dimensions, earlier work of Buchta was extended by
Bárány and Füredi [BáF88], who proved that
Eψn(d) (B d , n(d)) → 1
Eψm(d) (B d , m(d)) → 0
if
n(d) = 2d/2 d− ,
if
m(d) = 2d/2 d(3/4)+
Discrete aspects of stochastic geometry
5
when d → ∞, for every fixed  > 0. The authors also investigated k-neighborliness
of the convex hull.
We turn to random variables ϕ(K, n) connected with intrinsic volumes and
face numbers. First we mention the rare instances where information on the whole
distribution is available. Some special results for d = 2 due to Alagar, Reed,
and Henze are quoted in [Sch88, Section 4]. For example, Henze showed that the
distribution function FK of V2 (K, 3) for a convex body K ⊂ R2 satisfies FT ≤
FK ≤ FE , where T is a triangle and E is an ellipse, provided that K, T, E have the
same area. Results on the distribution of Vr (B d , r + 1) for r = 1, . . . , d are listed
in a more general context in Section 12.1.5.
In the plane, a few remarkable central limit type theorems have been obtained.
For a convex polygon P ⊂ R2 with r vertices, Groeneboom [Gro88] proved that
D
f0 (P, n) − 23 r log n D
q
→ N (0, 1)
10
n
r
log
27
for n → ∞, where → denotes convergence in distribution and N (0, 1) is the standard normal distribution. From this, Massé [Mas00] deduced that
lim
n→∞
3f0 (P, n)
=1
2r log n
in probability.
For the circular disk, Groeneboom showed
f0 (B 2 , n) − 2πc1 n1/3 D
p
→ N (0, 1)
2πc2 n1/3
2 1/3
with c1 = ( 3π
) Γ( 53 ) ≈ 0.53846 and c2 given by an integral which was evaluated
numerically. For a polygon P with r vertices, a result of Cabo and Groeneboom
[CaG94], in a version suggested by Buchta [Buc02], says that
V2 (P )−1 D2 (P, n) − 23 r logn n D
q
→ N (0, 1).
28 log n
r
27
n2
For the circular disk, a result of Hsing [Hsi94] was made more explicit by Buchta
[Buc02] and now gives, as n → ∞, π −2 varD2 (B 2 , n) ∼ 2π( 31 c1 + c2 )n−5/3 and
π −1 D2 (B 2 , n) − 2πc1 n−2/3 D
q
→ N (0, 1).
2π( 13 c1 + c2 )n−5/3
A thorough study of the asymptotic properties of D2 (µ, n) and D1 (µ, n) was
presented by Bräker and Hsing [BH98], for rather general distributions µ (including
the uniform distribution) concentrated on a convex body K in the plane, where K
is either sufficiently smooth and of positive curvature or a polygon. Bräker, Hsing
and Bingham [BHB98] investigated the asymptotic distribution of the Hausdorff
distance between a planar convex body K (either smooth or a polygon) and the
convex hull of n i.i.d. uniform points in K.
Küfer [Küf94] studied the asymptotic behavior of Dd (B d , n) and showed, in
particular, that its variance is at most of order n−(d+3)/(d+1) , as n → ∞.
Most of the known results about the random variables ϕ(K, n) concern their
expectations. Explicit formulae for E ϕ(K, n) for convex bodies K ⊂ Rd and arbitrary n ≥ d + 1 are known in the cases listed in Table 12.1.2.
6
R. Schneider
TABLE 12.1.2
DIMENSION d
2
2
2
3
≥2
≥2
Expected value of ϕ(K, n).
CONVEX BODY K
FUNCTIONAL ϕ
SOURCES
polygon
polygon
ellipse
ellipsoid
ball
ball
V2
f0
V2
V3
S, mean width, fd−1
Vd
Buchta [Buc84a]
Buchta and Reitzner [BuR97a]
Buchta [Buc84b]
Buchta [Buc84b]
Buchta and Müller [BuM84]
Affentranger [Aff88]
Affentranger’s result is given in the form of an integral, which can be evaluated
for given d and n; it implies the corresponding result for ellipsoids.
A well-known problem, popularized by Klee, is the explicit determination of
EVd (T d , d + 1) for a d-simplex T d . Klee’s opinion that EV3 (T 3 , 4) “might yield to
brute force” was justified. The result
EV3 (T 3 , 4) =
13
π2
−
= 0.0173982 . . .
720 15015
(12.1.4)
was announced by Buchta and Reitzner [BuR92], as well as a more general formula for EV3 (T 3 , n). Independently, (12.1.4) was established by Mannion [Man94],
who made heavy use of computer algebra. Finally, Buchta and Reitzner [BuR01]
published a readable version of their admirable proof for the formula
EV3 (T 3 , n) = pn − π 2 rn ,
with explicitly given rational numbers pn , rn .
If explicit formulae for E ϕ(K, n) are not available, one can try to obtain inequalities or asymptotic expansions for increasing n. For EVd (K, n), the following
estimates are known. The quotient EVd (K, n)/Vd (K), for n ≥ d + 1, is minimal
for ellipsoids (Groemer). The conjecture that it is maximal for simplices is only
proved for d = 2. (References for these and related results are given in the survey
part of [BaS95].) If the convex body K ⊂ Rd is not a simplex, then the quotient
EVd (K, n)/Vd (K) is strictly less than its value for a simplex, for all n ≥ n0 (K),
see [BáB93]. If Vd (K) = 1 and f is a continuous strictly increasing function, the
expectation Ef (Vj (K, n)) is minimal if K is a ball; this was proved by Hartzoulaki
and Paouris [HaP01].
We turn to asymptotic results for expectations. Buchta [Buc84c] considered
the perimeter and proved for plane polygons P that
“−1/2
’
n
+ o(n−1 )
ED1 (P, n) = c(P )
V2 (P )
for any fixed  > 0, where the constant c(P ) is given explicitly in terms of the
angles of P . For a convex polygon P with r vertices (and area one), Buchta and
Reitzner [BuR97a] obtained
Ef0 (P, n) =
2r
c1 (P ) c2 (P )
log n + c0 (P ) +
+
+ ...
3
n
n2
as n → ∞, with explicit constants ci (K); this strengthens a result of Rényi and
Sulanke. Further work of the latter authors for the plane is described in [Sch88,
Discrete aspects of stochastic geometry
7
Section 5], as well as some particular results for Rd , in part superseded by the
following ones. For d-dimensional polytopes P , Bárány and Buchta [BáB93] were
able to show that
Ef0 (P, n) =
T (P )
logd−1 n + O(logd−2 n log log n),
(d + 1)d−1 (d − 1)!
(12.1.5)
where T (P ) denotes the number of chains F0 ⊂ F1 ⊂ . . . ⊂ Fd−1 where Fi is an
i-dimensional face of P . They establish a corresponding relation for the volume,
from which (12.1.5) follows by (12.1.3). This work was the culmination of a series
of papers by other authors, among them Affentranger and Wieacker [AfW91], who
settled the case of simple polytopes, which is applied in [BáB93]. Bárány and
Buchta mention that their methods permit one to extend (12.1.5) to Efk (P, n) for
k = 0, . . . , d − 1, with the denominator replaced by a constant depending on d and
k.
For convex bodies K ⊂ Rd with a boundary of class C 3 and positive GaussKronecker curvature κ, Bárány [Bár92] obtained relations of the form
(j,d)
EDj (K, n) = c2
(K)n−2/(d+1) + O(n−3/(d+1) log2 n)
(12.1.6)
(1,d)
for j = 1, . . . , d. For j = 1, such a result (with explicit c2 ) was obtained earlier by
Schneider and Wieacker [ScWi80]. For simplicity, one can assume that Vd (K) = 1.
Then, for j = d, the coefficient is given by
Z
(d,d)
(d,d)
κ1/(d+1) dS
c2 (K) = c2
∂K
(d,d)
and thus is a constant multiple (c2
area of K. The limit relation
depending only on d) of the affine surface
(d,d)
lim n2/(d+1) EDd (K, n) = c2
n→∞
Z
κ1/(d+1) dS
∂K
was extended by Schütt [Schü94] to arbitrary convex bodies (of volume one),
with the Gauss-Kronecker curvature generalized accordingly. The other coefficients
(j,d)
c2 (K) in (12.1.6) are given by
Z
(j,d)
(j,d)
κ1/(d+1) Hd−j dS,
c2 (K) = c2
∂K
where Hd−j denotes the (d − j)th normalized elementary symmetric function of the
(j,d)
principal curvatures of ∂K and c2
depends only on j and d. These values were
given by Reitzner [Rei01b], thus correcting the coefficients shown in [Bár92].
Under stronger differentiability assumptions, more precise asymptotic expansions are possible. If K has a boundary of class C k+3 , k ≥ 2, and positive curvature
(and is of volume one), then
EDj (K, n)
(j,d)
= c2
(j,d)
(K)n−2/(d+1) + . . . + ck
(K)n−k/(d+1) + O(n−(k+1)/(d+1) )
8
R. Schneider
as n → ∞, where additional information on the coefficients is available. This was
proved by Reitzner [Rei01b] (for d = j = 2, see also Reitzner [Rei01a]). Under
the same assumptions on K, Gruber [Gru96] had obtained earlier an analogous
asymptotic expansion for η(C) − Eη(C, n), where η(C) is the value of the support
function of the convex body C at a given vector u ∈ S d−1 .
For general convex bodies K, the approximation behavior is typically irregular,
hence the main interest will be in sharp estimates. A first result concerns V1
(essentially the mean width). For a convex body K ⊂ Rd , Schneider [Sch87] showed
the existence of positive constants a1 (K), a2 (K) such that
a1 (K)n−2/(d+1) < ED1 (K, n) < a2 (K)n−1/d
(12.1.7)
for n ∈ N. Smooth bodies (left) and polytopes (right) show that the orders are best
possible.
For general convex bodies K, a powerful method for investigating the polytopes
Kn := conv{X1 , . . . , Xn }, for i.i.d. uniform points X1 , . . . , Xn in K, was invented
by Bárány and Larman [BáL88]. For K of volume one and for sufficiently small
t > 0, they introduced the floating body K[t] := {x ∈ K | Vd (K ∩ H) ≥ t for every
halfspace H with x ∈ H}. Their main result says that Kn and K[1/n] approximate
K of the same order and that K \ Kn is close to K \ K[1/n] in a precise sense.
From this, several results on the expectations Eϕ(K, n) for various ϕ were obtained
by Bárány and Larman [BáL88], Bárány [Bár89], for example
c1 (d)(log n)d−1 < Efj (K, n) < c2 (d)n(d−1)/(d+1)
(12.1.8)
for j ∈ {0, . . . , d} with positive constants ci (d) (the orders are best possible), and
by Bárány and Vitale [BáV93].
The inequalities (12.1.7) show that for general K the approximation, measured
in terms of D1 (K, ·), is not worse than for polytopes and not better than for smooth
bodies. For approximation measured by Dd (K, ·), the class of polytopes and the
class of smooth bodies interchange their roles, since
b1 (K)n−1 (log n)d−1 < EDd (K, n) < b2 (K)n−2/(d+1) ,
as follows from [BáL88] (or (12.1.8) for j = 0 and (12.1.3)). This observation lends
additional interest to Problem 12.1.3.
OPEN PROBLEMS
PROBLEM 12.1.1 (Valtr [Val96])
Is it true, for a convex body K ⊂ R2 and for n ≥ 4, that Eψn (K, n), the probability
that n uniform i.i.d. points in K are in convex position, is minimal if K is a triangle
and maximal if K is an ellipse?
PROBLEM 12.1.2
For a convex body K ⊂ Rd with a boundary of class C 3 and positive GaussKronecker curvature κ, and for the numbers of k-faces one expects that
’
“(d−1)/(d+1)
Z
n
1/(d+1)
κ
dS
(1 + o(1)) (12.1.9)
Efk (K, n) = b(d, k)
Vd (K)
∂K
Discrete aspects of stochastic geometry
9
with a constant b(d, k). For k = 0, this follows from (12.1.6); for k = d − 1 (which
implies the case k = d − 2) the result goes back to Raynaud and Wieacker; see
[Bár92] and [Sch88, p. 222] for references.
PROBLEM 12.1.3 (Bárány [Bár89])
Is it true for a general convex body K ⊂ Rd that the surface area satisfies
c1 (K)n−1/2 < EDd−1 (K, n) < c2 (K)n−2/(d+1)
with positive constants c1 (K), c2 (K)?
12.1.4 RANDOM POINTS ON CONVEX SURFACES
If µ is a probability distribution on the boundary ∂K of a convex body K and
µ has a density h with respect to the area measure of ∂K, we write ϕ(µ, n) =:
ϕ(∂K, h, n) and Dj (∂K, h, n) := Vj (K) − Vj (∂K, h, n). Some references concerning
Eϕ(∂K, h, n) are given in [Sch88, p. 224]. Most of them are superseded by an
investigation of Reitzner [Rei02a]. For K with a boundary of class C 2 and positive
Gauss curvature, j ∈ {1, . . . , d} and continuous h > 0, he showed that
Z
(j,d)
EDj (∂K, h, n) = b2
h−2/(d−1) κ1/(d−1) Hd−j dS · n−2/(d−1) + o(n−2/(d−1) )
∂K
as n → ∞. Under stronger differentiability assumptions on K and h, an asymptotic
expansion with more terms was established. Similar results for support functions
were obtained earlier by Gruber [Gru96].
For j = d, there is an asymptotic relation for general convex bodies K satisfying
only a weak regularity assumption. In a long and intricate proof, Schütt and Werner
[ScWe01] proved that
Z
(d,d)
lim n2/(d−1) EDd (∂K, h, n) = b2
h−2/(d−1) κ1/(d−1) dS,
n→∞
∂K
provided that the lower and upper curvatures of K are between two fixed positive
and finite bounds.
Let (Xk )k∈N be an i.i.d. sequence of uniform random points on the boundary ∂K of a convex body K. Let δ denote the Hausdorff metric. Dümbgen and
Walther [DüW96] showed that δ(K, conv{X1 , . . . , Xn }) is almost surely of order
O((log n/n)1/(d−1) ) for general K, and of order O((log n/n)2/(d−1) ) under a smoothness assumption.
OPEN PROBLEM
PROBLEM 12.1.4
Let K ⊂ Rd be a convex body with a boundary of class C 3 and positive GaussKronecker curvature κ. Let (Xk )k∈N be an i.i.d. sequence of random points in ∂K,
the distribution of which has a continuous positive density h with respect to the area
measure. We conjecture that
’
“2/(d−1)
’
√ “2/(d−1)
1
1
κ
n
δ(K, conv {X1 , . . . , Xn }) =
max
lim
n→∞ log n
2 κd−1
h
10
R. Schneider
with probability one. For d = 2, this is true, and similar results hold with the
Hausdorff distance replaced by area or perimeter difference; this was proved by
Schneider [Sch88]. For d > 2 and with convergence in probability instead of almost
sure convergence, the result was proved by Glasauer and Schneider [GlS94].
12.1.5 CONVEX HULLS FOR OTHER DISTRIBUTIONS
Convex hulls of i.i.d. random points have been investigated for each of the distributions listed in Table 12.1.1, and occasionally for more general ones. The following
setup has been studied repeatedly. For 0 ≤ p ≤ r + 1 ≤ d − 1, one considers r + 1
independent random points, of which the first p are uniform in the ball B d and the
last r + 1 − p are uniform on the boundary sphere S d−1 . Precise information on
the moments and the distribution of the r-dimensional volume of the convex hull is
available; see the references in [Sch88, pp. 219, 224] and the work of Affentranger
[Aff88].
Among spherically symmetric distributions, the beta distributions are particularly tractable. For these, again, the r-dimensional volume of the convex hull of
r + 1 i.i.d. random points has frequently been studied. We refer to the references
given in [Sch88] and Chu [Chu93]. Affentranger [Aff91] determined the asymptotic
behavior, as n → ∞, of the expectation EVj (µ, n), where µ is either the beta type-1
distribution, the uniform distribution in B d , or the standard normal distribution
in Rd . Also the asymptotic behaviour of Efd−1 (µ, n) was found for these cases.
Further information is contained in the book of Mathai [Mat99].
For normally distributed points in the plane, Hueter [Hue94] proved central
limit type results for the number of vertices, the perimeter, and the area of the
convex hull. For d ≥ 2, she obtained in [Hue99] a central limit theorem for f0 (µ, n),
for a class of spherically symmetric distributions µ in Rd including the normal
family. For the normal distribution µ2 in the plane, Massé [Mas00] derived from
[Hue94] that
f0 (µ2 , n)
lim √
=1
in probability.
n→∞
8π log n
For the expectations Efk (µd , n) (k ∈ {0, . . . , d − 1}), where µd is the standard
normal distribution in Rd , one knows that
’
“
2d
d
βk,d−1 (π log n)(d−1)/2
Efk (µd , n) ∼ √
(12.1.10)
k
+
1
d
as n → ∞, where βk,d−1 is the interior angle of the regular (d − 1)-dimensional
simplex at one of its k-dimensional faces. This follows from [AfS92], where the
Grassmann approach was used, due to the equivalence of [BaV94] explained in
Section 12.1.2. For the Grassmann approach, Vershik and Sporyshev [VeS92] have
made a careful study of the asymptotic behavior of the number of k-faces, if k and
the dimension d grow linearly with the number n.
Relation (12.1.10) describes also the asymptotic behavior of the number of
k-faces of the orthogonal projection of a regular (n − 1)-simplex onto a randomly
chosen isotropic d-subspace. In a similar investigation, Böröczky and Henk [BöH99]
replaced the regular simplex by the regular crosspolytope and found, surprisingly,
the same asymptotic behavior.
For more general spherically symmetric distributions µ, the asymptotic behavior of the random variables ϕ(µ, n) will essentially depend on the tail behavior
Discrete aspects of stochastic geometry
11
of the distribution. Extending work of Carnal (1970), Dwyer [Dwy91] obtained
asymptotic estimates for Ef0 (µ, n), Efd−1 (µ, n), EVd (µ, n), and ES(µ, n). Devroye
[Dev91] showed that for any monotone sequence ωn ↑ ∞ and for every  > 0, there
is a radially symmetric distribution µ in the plane for which Ef0 (µ, n) ≥ n/ωn
infinitely often and Ef0 (µ, n) ≤ 4 +  infinitely often. For ωn ↑ ∞ strictly increasing and satisfying ωn ≤ n, Massé [Mas99] constructed a distribution µ in the
plane such that the variance satisfies varf0 (µ, n) ≥ n2 /ωn infinitely often. Aldous
et al. [AlFGP91] considered an i.i.d. sequence (Xk )k∈N in R2 with a spherically
symmetric (or more general) distribution. Under an assumption of slowly varying tail, they determined a limiting distribution for f0 (µ, n). Massé [Mas00] constructed a distribution µ in the plane for which Ef0 (µ, n) → ∞ for n → ∞, but
(f0 (µ, n)/Ef0 (µ, n))n∈N does not converge to 1 in probability.
12.2 RANDOM POINTS – OTHER ASPECTS
For a finite set of points, the relative position of its elements may be viewed under
various geometric and combinatorial aspects. For randomly generated point sets,
the probabilities of particular configurations may be of interest, but are in general
hard to obtain. We list some contributions to problems of this type.
For infinite sets of points in the whole space, the natural generalization of i.i.d.
points in a compact domain are homogeneous Poisson processes.
12.2.1 GEOMETRIC CONFIGURATIONS
Bokowski et al. [BoRS92] made a simulation study to estimate the probabilities of
certain order types, using the Grassmann approach.
Related to k-sets (see Chapter 1 of this Handbook) is the following investigation
of Bárány and Steiger [BáS94]. If X is a set of n points in general position in Rd ,
a subset S ⊂ X of d points is called a k-simplex if X has exactly k points on
one side of the affine hull of S. The authors study Ed (k, n), the expected number
of k-simplices for n i.i.d. random points. For continuous spherically symmetric
distributions they show that
Ed (k, n) ≤ c(d)nd−1 .
Further results concern the uniform distribution in a convex body in R2 .
For a given distribution µ on R2 , let P1 , . . . , Pj , Q1 , . . . , Qk be i.i.d. points
distributed according to µ. Let pjk (µ) be the probability that the convex hull of
P1 , . . . , Pj is disjoint from the convex hull of Q1 , . . . , Qk . Continuing earlier work of
L.C.G. Rogers and of Buchta, Buchta and Reitzner [BuR97a] investigated pjk (µ).
For the uniform distribution µ in a convex domain K, they connected pjk (µ) to
equiaffine inner parallel curves of K, found an explicit representation in the case of
polygons and proved, among other results, that
√
pnn (µ)
8 π
lim
≥
,
n→∞ n3/2 4−n
3
with equality if K is centrally symmetric. The investigation was continued by
Buchta and Reitzner [BuR97b].
12
R. Schneider
Various elementary geometric questions can be asked, even about a small number of random points. For example, if three uniform i.i.d. points in a convex body
K are given, what is the probability that the triangle formed by them is obtuse, or
what is the probability that the circle (almost surely) determined by these points
is contained in K? Known results on probabilities of these types are listed in
[BaS95]. The following result is due to Affentranger. The probability that the
sphere spanned (almost surely) by d + 1 i.i.d. uniform random points in a convex
body K is entirely contained in K attains its maximum precisely if K is a ball. In
[BaS95] it is shown that the probability that the circumball of n ≥ 2 i.i.d. uniform
points in K is contained in K is maximal if and only if K is a ball. The value of
this maximum is n/(2n − 1) if d = 2, but is unknown for d > 2.
Many special problems are treated, and references are given, in the book of
Mathai [Mat99].
12.2.2 SHAPE
Two subsets of Rd may be said to have the same shape if they differ only by a
similarity. D.G. Kendall’s theory of shape yields natural probability distributions
on shapes of labeled n-tuples of points in Rd . The possible shapes of such n-tuples of
points (not all coincident) can canonically be put in one-to-one correspondence with
points of a certain topological space, and the resulting “shape spaces” carry natural
probability measures. For this extensive theory and its statistical applications, we
refer to the survey given by Kendall [Ken89] and to the book of Kendall et al.
[KeBCL99].
A different approach to more general notions of shape and probability distributions for them is followed by Ambartzumian [Amb90]. He uses factorization of
products of invariant measures to obtain corresponding probability densities, for
example, for the affine shape of a tetrad of points in the plane.
12.2.3 POINT PROCESSES
The investigations described so far concerned finite systems of random points. For
randomly generated infinite discrete point sets, suitable models are provided by
stochastic point processes.
GLOSSARY
Locally finite: M ⊂ Rd is locally finite if card (M ∩ B) < ∞ for every compact
set B ⊂ Rd .
M: The set of all locally finite subsets of Rd .
M: The smallest σ-algebra on M for which every function M 7→ card (M ∩ B) is
measurable, where B ⊂ Rd is a Borel set.
(Simple) point process X on Rd : A measurable map X from some probability
space (Ω, A, P ) into (M, M).
Distribution of X: The image measure PX of P under X.
Intensity measure Λ of X : Λ(B) = E card (X ∩ B), for Borel sets B ⊂ Rd .
Stationary (or homogeneous): X is a stationary point process if the distribution PX is invariant under translations.
Discrete aspects of stochastic geometry
13
The point process X on Rd , with intensity measure Λ (assumed to be finite
on compact sets), is a Poisson process if, for any finitely many pairwise disjoint
Borel sets B1 , . . . , Bk , the random variables card (X ∩ B1 ), . . . , card (X ∩ Bk ) are
independent and Poisson distributed. Thus, a Poisson point process X satisfies
Prob{card (X ∩ B) = k} = e−Λ(B)
Λ(B)k
k!
for k ∈ N 0 and every Borel set B. If it is stationary, then the intensity measure
Λ is γ times the Lebesgue measure, and the number γ is called the intensity of
X. Let X be a stationary Poisson process and C ⊂ Rd a compact set, and let
k ∈ N 0 . Under the condition that exactly k points of the process fall into C, these
points are equivalent to k i.i.d. uniform points in C. This fact clearly illustrates the
geometric significance of stationary Poisson point processes, as does the following.
Consider n i.i.d. uniform points in the ball rB d . The Poisson process with intensity
measure equal to the Lebesgue measure can be considered as the limit process that
is obtained if n and r tend to infinity in such a way that n/Vd (rB d ) → 1.
A detailed study of geometric properties of stationary Poisson processes in the
plane was made by Miles [Mil70].
For much of the theory of point processes, the underlying space Rd can be
replaced by a locally compact topological space Σ with a countable base. Of importance for stochastic geometry are, in particular, the cases where Σ is the space
of r-flats in Rd (see Section 12.3.3) or the space of convex bodies in Rd .
12.3 RANDOM FLATS
Next to random points, randomly generated r-dimensional flats in Rd are the objects of study in stochastic geometry that are particularly close to discrete geometry.
Like convex hulls of random points, intersections of random halfspaces yield random polytopes in a natural way. Random flats through convex bodies as well as
infinite arrangements of random hyperplanes give rise to a variety of questions.
12.3.1 RANDOM HYPERPLANES AND HALFSPACES
Intersections of random halfspaces appear as solution sets of systems of linear inequalities with random coefficients. Therefore, such random polyhedra play a role
in the average case analysis of linear programming algorithms (see the book by
Borgwardt [Bor87] and its bibliography). Under various assumptions on the distribution of the coefficients, one has information on the expected number of vertices
of the solution sets. Extending earlier work of Prékopa, Buchta [Buc87a] obtained
several estimates, of which the following is an example.
Let E(v)Pbe the expected number of vertices of the polyhedron given by the
n
inequalities
i=1 aij xj ≤ b (i = 1, . . . , m), xj ≥ 0 (j = 1, . . . , n). If the coefficients
aij are nonnegative and distributed independently, continuously, and symmetrically
with respect to the same number c > 0, then
’ “
’
“
1
m
n
n
E(v) = m−1
+ m−1
+ O(nm−2 )
2
2
m
m−1
for n → ∞. Buchta [Buc87b] also has formulae and estimates for E(v) in the
14
R. Schneider
Pn
case of the polyhedron given by j=1 aij xj ≤ 1 (i = 1, . . . , m), where the points
(ai1 , . . . , ain ) (i = 1, . . . , m) are i.i.d. uniform on the sphere S d−1 .
In a certain duality to convex hulls of random points in a convex body, one
may consider intersections of halfspaces containing a convex body. Let K ⊂ Rd be
a convex body with a boundary of class C 3 and with positive Gauss curvature κ;
−
:= {x ∈
suppose that 0 ∈ int K and let r > 0. Call a random closed halfspace Hu,t
d
d−1
R | hx, ui ≤ t} with u ∈ S
and t > 0 “(K, r)-adapted” if the unit normal vector
u is uniform on S d−1 and the distance t is independent of u and is, for given u,
−
contains K but not rB d . Let EṼd (K, n) be
uniform in the interval for which Hu,t
the expected volume of the intersection of rB d with n i.i.d. (K, r)-adapted random
halfspaces. Then Kaltenbach [Kal90] proved that
EṼd (K, n) − Vd (K)
=
c1 (d)
Z
κ
1/(d+1)
∂K
−3/(d+1)
+O(n
dS
’
n
V1 (rB d ) − V1 (K)
“−2/(d+1)
+
) + O(rd (1 − )n )
for n → ∞, where 0 <  < 1 is fixed.
Let X1 , . . . , Xn be i.i.d. random points on the boundary of a smooth convex body K. Let K(n) be the intersection of the supporting halfspaces of K at
X1 , . . . , Xn (intersected with some fixed large cube, to make it bounded), and put
D(j) (K, n) := Vj (K(n) ) − Vj (K). Under the assumption that the distribution of the
Xi has a positive density h and that K and h are sufficiently smooth, Böröczky and
Reitzner [BöR02] have obtained asymptotic expansions, as n → ∞, for ED(j) (K, n)
in the cases j = d, d − 1, 1.
Let (Xi )i∈N be a sequence of i.i.d. random points on ∂K, and let K(n) and h
be defined as above. If ∂K is of class C 2 and positive curvature and h is positive
and continuous, one may ask whether n2/(d−1) D(j) (K, n) converges almost surely
to a positive constant, if n → ∞. For d = 2 and j = 1, 2, this was shown by
Schneider [Sch88]. Reitzner [Rei02b] was able to prove such a result for d ≥ 2 and
j = d. He deduced that random approximation, in this sense, is very close to best
approximation.
12.3.2 RANDOM FLATS THROUGH CONVEX BODIES
The notion of a uniform random point in a convex body K in Rd is extended by that
of a uniform random r-flat through K. Let Erd be the space of r-dimensional affine
subspaces of Rd with the usual topology and Borel structure (r ∈ {0, . . . , d − 1}).
A random r-flat is a measurable map from some probability space into Erd . It is a
uniform (isotropic uniform) random r-flat through K if its distribution can be
obtained from a translation invariant (resp. rigid-motion invariant) measure on Erd ,
by restricting it to the r-flats meeting K and normalizing to a probability measure.
(For details, see [WeW93, Section 2] and [ScW00, Chapter 4].)
A random r-flat E (uniform or not) through K generates the random secant
E ∩ K, which has often been studied, particularly for r = 1. References are in
[ScW92, Chapter 6] and [ScWi93, Section 7]. Finitely many i.i.d. random flats
through K lead to combinatorial questions. Associated random variables, such as
the number of intersection points inside K if d = 2 and r = 1, are hard to attack;
for work of Sulanke (1965) and Gates (1984) see [ScWi93]. Of special interest is the
Discrete aspects of stochastic geometry
15
case of i ≤ d i.i.d. uniform hyperplanes H1 , . . . , Hi through a convex body K ⊂ Rd .
Let pi (K) denote the probability that the intersection H1 ∩. . .∩Hi also meets K. In
some special cases, the maximum of this probability (which depends on K and on
the distribution of the hyperplanes) is known, but not in general. References for this
and related problems and a conjecture are found in [BaS95]. If N > d i.i.d. uniform
hyperplanes through K are given, they give rise to a random cell decomposition of
int K. For k ∈ {0, . . . , d}, the expected number, Eνk , of k-dimensional cells of this
decomposition is given by
“’ “
’
d
X
N
i
pi (K),
Eνk =
d−k
i
i=d−k
with pi (K) as defined above (Schneider [Sch82]). If the hyperplanes are isotropic
uniform, then
“’ “
’
d
X
N i!κi Vi (K)
i
Eνk =
.
i 2i V1n (K)
d−k
i=d−k
OPEN PROBLEM
PROBLEM 12.3.1
For i ≤ d i.i.d. uniform random hyperplanes through a convex body K, find the
sharp upper bound for the probability that their intersection also intersects K.
12.3.3 POISSON FLATS
A suitable model for infinite discrete random arrangements of r-flats in Rd is provided by a point process in the space Erd . Stationary Poisson processes are the
simplest and geometrically most interesting examples. Basic work was done by
Miles [Mil71a] and Matheron [Math75]. For a hyperplane process, an ith intersection density ∆i can be defined, in such a way that, for a Borel set A ⊂ Rd ,
the expectation of the total i-dimensional volume inside A, of the intersections of
any d − i hyperplanes of the process, is given by ∆i times the Lebesgue measure
of A. Given the intensity ∆d−1 , the maximal ith intersection density ∆i (for an
i ∈ {0, . . . , d − 2}) is achieved if the process is isotropic (its distribution is rigidmotion invariant); this result is due to Thomas (1984, see [ScW00, Section 4.5]).
(Result and method were carried over to deterministic discrete hyperplane systems
by Schneider [Sch95].) Similar questions can be asked for stationary Poisson r-flats
with r < d − 1, for example for 2r ≥ d and intersections of any two r-flats. Here
nonisotropic extremal cases occur, such as in the case r = 2, d = 4 solved by Mecke
[Mec88]. Various other cases have been treated; see Mecke [Mec91], Keutel [Keu91],
and the references given there.
12.4 RANDOM CONGRUENT COPIES
The following is a typical question on randomly moving sets. Let K0 , K ⊂ Rd be
16
R. Schneider
given convex bodies. An isotropic random congruent copy of K meeting K0 is of
the form gK, where g is a random element of the motion group Gd of Rd , and the
distribution of g is obtained from the Haar measure on Gd by restricting it to the
set {g ∈ Gd | K0 ∩ gK 6= ∅} and normalizing. Let K1 , . . . , Kn be convex bodies, let
gi Ki be an isotropic random copy of Ki meeting K0 , and suppose that g1 , . . . , gn
are stochastically independent. What is the probability that the random bodies
g1 K1 , . . . , gn Kn have a common point inside K0 ? This question and similar ones
can be given explicit answers by means of integral geometry. We refer to the books
of Santaló [San76] and of Schneider and Weil [ScW92].
12.5 RANDOM MOSAICS
By a tessellation of Rd , or a mosaic in Rd , we understand a collection of ddimensional polytopes such that their union is Rd , the intersection of any two of
the polytopes is either empty or a face of each of them, and any bounded set meets
only finitely many of the polytopes. A random mosaic can be modeled by a
point process in the space of convex polytopes, such that the properties above are
satisfied almost surely. General references are [Møl89], [MeSSW90, Chapter 3],
[WeW93, Section 7], [StKM95, Chapter 10], and [ScW00, Chapter 6].
NOTATION
stationary random mosaic in Rd
process of its k-dimensional faces
density of the jth intrinsic volume of the polytopes in X (k)
(k)
= d0 , k-face intensity of X
typical k-face of X
expected number of elements of X (k) that are typically incident
with a j-face of X
X
X (k)
(k)
dj
γ (k)
Z (k)
njk
Under a natural assumption on the stationary random mosaic X, the notions of
(k)
‘density’ and ‘typical’ exist with a precise meaning. The density dj is then the
intensity γ (k) times the expectation of the jth intrinsic volume of the typical k-face
Z (k) . Here, we can only convey the intuitive idea that one averages over expanding
bounded regions of the mosaic and performs a limit procedure. Exact definitions
are found in [ScW00]; to Chapter 6 of that book we also refer for the results listed
below and all the pertaining references.
12.5.1 GENERAL MOSAICS
For arbitrary stationary random mosaics, there is a number of identities between
averages of combinatorial quantities. Basic examples are:
j
X
(−1)k njk = 1,
k=0
d
X
k=j
(−1)d−k njk = 1,
γ (j) njk = γ (k) nkj ,
Discrete aspects of stochastic geometry
d
X
(i)
(−1)i dj = 0,
especially
d
X
17
(−1)i γ (i) = 0.
i=0
i=j
If the random mosaic X is normal, meaning that every k-face is contained in exactly
d − k + 1 d-polytopes of X (k = 0, . . . , d − 1), then
k
(1 − (−1) )γ
(k)
“
’
d + 1 − j (j)
γ .
=
(−1)
k−j
j=0
k−1
X
j
Lurking in the background are, of course, polytopal relations of Euler, DehnSommerville, and Gram.
12.5.2 HYPERPLANE TESSELLATIONS
A random mosaic X is called a stationary hyperplane tessellation if it is induced, in
the obvious way, by a stationary hyperplane process (as defined in Section 12.3.3).
Such random mosaics have special properties. Under an assumption of general
position (satisfied, for example, by Poisson hyperplane processes) one has, for 0 ≤
j ≤ k ≤ d,
“
’ “
’
d − j (j)
d (0)
(k)
dj ,
especially γ (k) =
γ ,
dj =
d−k
k
and
nkj = 2k−j
’ “
k
.
j
A stationary Poisson hyperplane process, satisfying a suitable assumption of nondegeneracy, induces a stationary random mosaic X in general position, called a
Poisson hyperplane mosaic. In the isotropic case (where the distributions are
invariant under rigid motions), X is completely determined by the intensity, γ̂, of
the underlying Poisson hyperplane process. In this case, one has
(k)
dj
=
’
d−j
d−k
“’ “
κd−j
d
d−1
γ̂ d−j ,
j dd−j κd−j−1
κj
d
in particular,
γ (k) =
’ “ d
d κd−1 d
γ̂ ,
k dd κd−1
d
EVj (Z (k) ) =
’ “’
“j
k
1
dκd
.
κd−1
κj γ̂ j
j
The, almost surely unique, d-polytope of X containing 0 is called the Poisson zerocell and denoted by Z0 . For a stationary Poisson hyperplane mosaic, the inequalities
EVd (Z0 ) ≥ d!κd
’
2κd−1
γ̂
dκd
“−d
and
2d ≤ Ef0 (Z0 ) ≤ 2−d d!κ2d
are valid. In the isotropic case, equality holds in the first and on the right-hand
side of the second inequality.
18
R. Schneider
For the typical cell Z (d) , the distribution of the inradius I (radius of the
largest contained ball) can be determined; it is given by Prob{I(Z (d) ) ≤ a} =
1 − exp (−2γ̂a) for a ≥ 0.
12.5.3 VORONOI AND DELAUNAY MOSAICS
A discrete point set in Rd induces a Voronoi and a Delaunay mosaic (see Chapter
23 for the definitions). Starting from a stationary Poisson point process X̃ in Rd ,
one obtains in this way a stationary Poisson-Voronoi mosaic and PoissonDelaunay mosaic. Both of these are completely determined by the intensity,
γ̃, of the underlying Poisson process X̃. For a Poisson-Voronoi mosaic and for
k ∈ {0, . . . , d}, one has
‘ €
 2
d−k+ kd €

d−k+1 d−k
Γ d −kd+k+1
Γ 1 + d2
Γ d − k + kd
2
d−k
2
π
2
(k)
dk =
γ̃ d .
€
€

€


d−k
2
d(d − k + 1)!
Γ d+1
Γ k+1
Γ d −kd+k
2
2
2
In particular, the vertex density is given by
 2 ‘"
 #d
€
d +1
d
Γ
d+1 d−1
2
Γ
1
+
2
2
π
€ 2
€ 
γ (0) = 2
γ̃.
d (d + 1) Γ d22
Γ d+1
2
For many other parameters, their explicit values in terms of γ̃ are known, especially
in small dimensions.
For a Poisson-Delaunay mosaic one can, in a certain sense, explicitly determine
the distribution of the typical d-cell and the moments of its volume.
12.6 SOURCES AND RELATED MATERIAL
SOURCES FOR STOCHASTIC GEOMETRY IN GENERAL
Stoyan, Kendall, and Mecke [StKM95]: A monograph on theoretical foundations
and applications of stochastic geometry.
Matheron [Math75]: A monograph on basic models of stochastic geometry and
applications of integral geometry.
Santaló [San76]: The classical work on integral geometry and its applications to
geometric probabilities.
Schneider and Weil [ScW00]: An introduction to the mathematical models of
stochastic geometry, with emphasis on the application of integral geometry and
functionals from convexity.
Kendall and Moran [KeM63]: A collection of problems on geometric probabilities.
Solomon [Sol78]: A selection of topics from geometric probability theory.
Mathai [Mat99]: A comprehensive collection of results on geometric probabilities,
in particular of those types where analytic calculations lead to explicit results.
Discrete aspects of stochastic geometry
19
Klain and Rota [KlR97]: An introduction to typical results of integral geometry,
their interpretations in terms of geometric probabilities, and counterparts of discrete
and combinatorial character.
Ambartzumian [Amb90]: Develops a special approach to stochastic geometry via
factorization of measures, with various applications.
Moran [Mor66], [Mor69], Little [Lit74], Baddeley [Bad77]: “Notes on recent research
in geometrical probability,” useful surveys with many references.
Baddeley [Bad82]: An introduction and reading list for stochastic geometry.
Baddeley [Bad84]: Connections of stochastic geometry with image analysis.
Weil and Wieacker [WeW93]: A comprehensive handbook article on stochastic
geometry.
RELATED CHAPTERS
Several topics are outside the scope of this chapter, although they could be subsumed under probabilistic aspects of discrete geometry. Among these are randomization and average-case analysis of geometric algorithms and the probabilistic analysis of optimization problems in Euclidean spaces. Two classical topics of discrete
geometry, namely packing and covering, were also excluded, for the reason that the
existing probabilistic results are in a spirit rather far from discrete geometry.
Chapters of this Handbook in which these and related topics are covered are:
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
Chapter
1:
2:
16:
18:
23:
40:
46:
Finite point configurations
Packing and covering
Basic properties of convex polytopes
Face numbers of polytopes and complexes
Voronoi diagrams and Delaunay triangulations
Randomization and derandomization
Mathematical programming
RELEVANT SURVEYS AND FURTHER SOURCES
Some of the topics treated have been the subjects of earlier surveys. The following
sources contain references to the excluded topics as well as to work within the scope
of this chapter.
Borgwardt [Bor87], [Bor99], Shamir [Sha93]: Information on the probabilistic analysis of linear programming algorithms under different model assumptions.
Dwyer [Dwy88] and later work: Contributions to the average-case analysis of geometric algorithms.
Hall [Hal88]: A monograph devoted to the probabilistic analysis of coverage problems.
Buchta [Buc85]: A survey on random polytopes.
Schneider [Sch88], Affentranger [Aff92]: Surveys on approximation of convex bodies
by random polytopes.
20
R. Schneider
Gruber [Gru97], Schütt [Schü02]: Surveys comparing best and random approximation of convex bodies by polytopes.
Bauer and Schneider [BaS95]: A collection of information on inequalities and extremum problems for geometric probabilities.
REFERENCES
[Aff88]
F. Affentranger. The expected volume of a random polytope in a ball. J. Microscopy,
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