Image Anal Stereol 2010;29:13-18
Review article
CONVEX BODIES AND GAUSSIAN PROCESSES
Richard A. Vitale
Department of Statistics, University of Connecticut, Storrs, CT USA 06269 e-mail: r.vitale@uconn.edu
(Accepted October 11, 2009)
ABSTRACT
For several decades, the topics of the title have had a fruitful interaction. This survey will describe some of these connections, including the GB/GC classification of convex bodies, Ito-Nisio singularities from a geometric viewpoint, Gaussian representation of intrinsic volumes, the Wills functional in a Gaussian context, and inequalities.
Keywords: convex body, Gaussian process, intrinsic volume, random convex body, strong law of large numbers for random convex bodies, Wills functional.
INTRODUCTION
For several decades, the topics of the title have had a fruitful interaction. This survey will describe some of these connections, including the GB/GC classification of convex bodies, Ito-Nisio singularities from a geometric viewpoint, Gaussian representation ofintrinsic volumes, the Wills functional in a Gaussian context, and inequalities. For fuller discussions and references, the interested reader is urged to consult the bibliography.
GEOMETRIC PRELIMINARIES AND NOTATION
The setting is either finite dimensions or infinite dimensions, that is, Rd or l2. Schneider (1993) gives an excellent treatment of the classical theory of convex bodies. The following items and notation will be appear:
•	Convex bodies K: compact, convex K, L,...
•	Scaling: 1K = {1x: x G K}.
•	Minkowski addition: K + L = {x+y: x G K, y G L}.
•	Closed unit ball: B, Bd .
•	l-parallel body: K + IB.
•	Support function: hK(x) = suptGKt>.
•	Hausdorff metric:
p(K, L) = inf{1 > 0: K C L + IB, L C K+IB}
= sup |hK(u) - hL(u)|.
iui=1
•	Norm: |K| = maxxGK = maxHuH=1 hK (u).
GAUSSIAN PROCESSES WITH ISONORMAL INDEXING
For background and references on Gaussian processes, one can consult, for example, Lifshits (1995) and Bogachev (1998). We assume throughout a sequence of independent standard (i.e., N(0,1)) Gaussian random variables:
For a convex body K c l2 and t G K, we consider the map
t ^ At = <t, z> = X ^iZi.
i=1
The image is an N(0, |t|2) variable, and the collection {X^t, t G K} is called an isonormall^-indexed Gaussian process in view of the isometric-isomorphism:
K ^^{^t, t G K} .
Specifically,
at + bt ^ a^t + b^i
t - tl2 = E (^t - Xt)2
Another key point is the identification
hK(Z) = sup<t, Z> = sup X^t.
t K	t K
LIMIT THEOREMS
Consider a random convex body X, which is a measurable map from a probability space to its space of values endowed with the Hausdorff metric:
X: {W, F, P}-^ (K, p).
If X is bounded in expected norm, E^Xy < then one has an expectation EX G K, which can be given implicitly in terms of its support function
hEX (■) = EhX (■).
There is a strong law of large numbers:
Theorem 1 (Artstein and Vitale, 1975) // X1,X2,...
a^re independent a^nd identically distributed r^andom convex bodies with E yX1 y < ¥, then
JT = X1 + X2 + ••• + Xn a-s\ EX] n	n	'
The formulation of an accompanying central limit theorem takes into account that there is no convenient notion of subtraction for convex bodies, and so the identification with support functions is used:
Theorem 2 (Weil, 198J2) // X1,X2,... a^re iid and E yXy2 < ¥, the^ ^/n [h^n(u) — hEX1 (u^ converges to a centered Gaussian process with inherited covariance function.
A different kind of limit theorem appears in Bonetti and Vitale (2000).
THE STEINER FORMULA AND INTRINSIC VOLUMES
The Steiner formula for the volume of the parallel body to a convex body in Fd is
vold(K+1B) = volj(Bj)1 Vd—j(K), J=o
where the constants Vj(K), J = 0,1,..., d are known as intrinsic volumes.
Following Vitale (1995), we give a derivation of the formula, which also serves to display the nature of the intrinsic volumes: consider iid isotropic line segments ,..., Ln, such that EL1 = Bd. By the strong law of large numbers,
(1/n)(Li + ••• + L„) ^ Bd
as n ^ and so
vold [K + (l/n) (Li + • • • + Ln)] ^ vold (K + IB^). For one line segment (i.e., n = 1), one has vold(K + IL1) = vold(K) +11L11 ■ vold—1 (nLxK),
where nL± signifies projection onto the subspace orthogonal to the one spanned by L1 . By induction,
vold [K +(X/n)(L1 + ••• + L„)] =
X (l/n)Svol | 5|(L5) vold— | SI HlxK) ,
0<|S|<d
where Ls = X^sLj. This can be re-expressed as
vold[K + (1/n) (L1 + ■■■ + Ln)] = X^J 1'Ujn,
J=0 n
where
1
Ujn = y-Y X volj (Ls) vold—j Hlx K
M ISi=j	V	s ,
has the form of a U-statistic. Then
vold (K + IBd) = lim vold [K + (1/n) (L1 + ■■■ + Ln)]
lim ^

jn
n j=0 nj = ^ ^J lim Ujn
j=0 J- n = ^ 11jcd—jEvold—j {nd—jK)
J=0 J-
= ^ volj^Bj)1J Vd—j(K), J=0
where Pj signifies projection onto a random subspace of dimension J and
Vj (K) =
d
vold (Bd)
\JJ volJ(BJ)vold—J(Bd—j)
EvolJ (njK).
(1)
A Gaussian version, shown below in Eq. 2, follows from noticing that, in Eq. 1, a key property is that, for an independent, random orthogonal O,
nJO=nJ.
n
It is also true that

where Z[ jdd] ,is a j x d matrix of independent N{0,1) variables. This can be used (Vitale, 2008) to show
Vj (K) =
{2K)j/2Evolj (Z[j,d]K)
(Bj)
Next we identify some of the intrinsic volumes:
V0(K) = 1
V1(K) = intrinsic width
(2)
Vj
^V2nEhK(Z) ^V2nEsup
re K
Vd-1 (K) = 1/2 ■ surface area of K Vd(K) = d-dimensional volume of K Vj(K)= 0 for j > d
/ n	\
bi] = ^ {bi1 - ai1) ■ ■ ■ {bij - Sij) V 1 J i1 <i2<-<ij
V1 (Bd) ^Vlpd.
EXTENSION OF INTRINSIC VOLUMES TO CONVEX BODIES IN I2
The extension of intrinsic volumes to convex bodies in Hilbert space and specifically to l2 was undertaken by Sudakov (1971) and Chevet (1976). To begin, let us identify the following collections:
convex bodies in Fd convex bodies in l2 finite-dimensional convex bodies in I2.
Kd
K
KFD
In view of the monotonicity of the intrinsic volumes under set inclusion, it is natural to extend them to infinite dimensional convex bodies as follows: for arbitrary K G K, define
Vj (K) = sup{ Vj (K) : I( C K, K G KFd } Kgb = {K G K : V1(K) < ^}.
GB stands for "Gaussian Bounded" (Dudley, 1967) and refers to the following identification.
The following also hold:
1.	Kfd C KGB C K .
2.	K G Kgb ^ Vj(K) < j = 2,3,...
3. K G Kgb ^ Vj(K) =
(2K)j/2 Evol
j! volj(Bj) ' where Z[ j¥] is a j x ¥ matrix of independent N(0,1) variables. Equivalently (Tsirel'son, 1985),
Vj (K) =
(2n)j/2 Evolj ({[A:!, a-2,..., Xj), t G K}
j! volj (Bj)
Some canonical cases are given in the next example.
Example Given a decreasing sequence of positive constants {an} and an orthonormal set {en, n = 1,2,...}, set
K = conv{anen, n = 1,2,...}.
Then
KG K K G KFD
K G Kgb
an i 0,
an = 0 eventually,
an = O
(log n)
-1/2
Example An example of an infinite-dimensional convex body that has no finite-dimensional analogue is as follows. Consider a map
f: [0,1] ^ H (Hilbert space)
that satisfies
1.	0 < X1 < X2 < X3 < X4 < 1 ^
[ f(x2) - f(x1)] ± [ f(x4) - f(x3)]
2.	y f(x2) - f(x1) y2 = |X2 - X11 for all 0 < X1 < X2 < 1. The associated Brownian Motion Body is defined to be
öönv{ f([0,1])}C H.
All Brownian motion bodies are the same, of course, up to an isometry. A particular realization in L2[0,1] is
{^: [0,1] ^ R1 | 0 < ^< 1, ^T}.
Theorem 3 K G Kgb
almost-surely bounded Gaussian process. That is P(sup^Ko Xt < ¥) = 1 for any denumerable subset K0 C K.
{^t, t G K} is an Theorem 4 (Gao and Vitale, 2001)
Vj (B^B) =
volj(Bj)
j = 1,2,...
SINGULARITIES
Although intrinsic volumes are defined, and finite, for all GB convex bodies, they are not continuous. That is, it is possible to have GB bodies with Kn ^ K, but Vj(Kn) Vj(K). In particular, one can have
Kn i{p}, but V1(Kn) y 0 = V1({p}).
This leads to the following definition.
Definition t* ^ K G KgB is a singularity of K if
V1(Kn B(t*, e)) y0 as e - 0.
Definition KgC = {K G KgB : K has no singularities}.
One has
KFD c KGC C KGB C K .
GC stands for "Gaussian Continuous" (Dudley, 1967), and the following gives the connection.
Theorem 5 K G KgC ^^ {X^t, t G K} is an almost-surely continuous Gaussian process. That is — t ^ — Xf almost-surely.
Example (continued)
K = öönv{anen, n = 1,2,...} G KGc
an = o
(log n)
-1/2
ITO-NISIO THEORY
Theorem 6 (Ito and Nisio, 1969) Suppose that t* G K G Kgb. The oscillation of X at t*, given by
0 < 2 ■ osc(t*) = lim
ejC
sup X^t — inf X^t
t GKnB(f*,e) tGKnB(t*-,s)
is almost surely constant. Further, osc(t*) > 0 ^^ t* is a singular^ity of K.
The following elaborates this observation.
Theorem 7 (Vitale, 2001) Suppose ^ha^tosc (t*) > 0. Then
1.	osc(t*)	iimey0 Vi(Kn B(tf, e)).
2.	For each J,
lim Vj(Kn B(t*, e)) > 0.
ei0
(3)
3. K n B(t*, 0+)

-B¥(f,osc(f)) in the
sense that for each J, the limit in Eq. 3 is equal to
1
B(t *, osc (t *))
lim Vj

/
A , , oscJ(t*) . (both being-^^^).
4. Define osc (K) = sup{Eosc (t*) : t* G K}. T^hen
(J + 1)Vj+i(K)
osc (K) = lim
J-
Vj (K)
THE WILLS FUNCTIONAL AND BOUNDS FOR GAUSSIAN PROCESSES
In the context of a question in lattice point enumeration, Wills (1973) defined the following functional. It has come to play an important role in the connection between the theories of convex bodies and Gaussian processes.
An alternate representation can be derived as follows: 1
W(K) = ^yRdP(dist(x, K) < A) dx,
where fA(1) = 1(1 > 0)Ae E 1(dist(x, K) < A) dx
— (1/2)12
1
(2n)d/2J^
1 e[ 1(dist(x, K) < A) dx
jRd
(2p)d/2 1
(2n)d/2
E vol(K+AB).
(2n)d/2
X voh(B)AiV,—(K) i=0
1 d
voli<Bi) [EA]Vd—(K)
1
(2p)d/2 Svoli(Bi)
(2n)i/2
voli(Bi)
Vd—i(K)
d
J?0 (2P)-J-/
Vj(K) .
2J
1
A second alternate representation proceeds as follows:
W{K) =
Rd (2p)d/2
1 e-5dist2(^-K) dx
I _1_e-2infteK W^dx
jRd (2p)d/2e	dX
(2p)d/ esuptGK [
= £esupteK[Xt-2 of].
= I esuPteK [<x,t>-1 WfW2]_1_
iRd	(2p)d/2
e-2WxW dx
One then has
PesuPtGK[Xt-1 o2] = Y _1_
J=0
(K) .
This can be extended by writing rK, r > 0, in place of K, which itself can be taken to be an element of Kgb:
^ / \ J EesupteK[rXt-^r2o2] = W	j yj(K) .
The Alexandrov-Fenchel inequality (Schneider, 1993) implies that
Vj(K) < "it V1J(K) = "it
VlnE sup Xf
reK
and hence
¥ /

VJ(K) < erEsupteKXt.
Thus we have shown the following:
Theorem 8 (Tsirel'son, 1985; Vitale, 1996, 2001 ) If
{X^t, t G K} is a mean-zero, bounded Gaussian process, then
Eesupt{Xt-(1/2)EXf} < eEsuptXt.
An immediate consequence is a deviation bound:
Theorem 9 (Pisier, 1986; Vitale, 1996,1999)
P(supXt - EsupXt > a) < exp[-(1/2)(a2/o2)]. (4)
tt
Proof: Set o2 = suptGK It is direct to show
Eer[suptGKXt-EsuptGKXd < e5r202 .
Then
P(sup Xt - E sup Xt > a) =
teK	teK
= P(r[sup Xt - E sup Xt] > ra)
teK	teK
= P(er[suptGKXt-EsuptGKXt] > era) < Eer[suptGKXt-EsuptGKXt]e-ra
<e2
2 r- o-ra
which is minimized at r = a/o2, and the assertion follows.
CONCLUSION
We have surveyed topics that combine elements of the theory of convex bodies and Gaussian processes.
ACKNOWLEDGMENT
An earlier version of this paper appeared in the accompanying Proceedings of the 10th European Congress of Stereology and Image Analysis (Milan, 2009).
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