CONVEX BODIES AND GAUSSIAN PROCESSES

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.


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 of intrinsic 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, I R d or ℓ 2 .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, . . .
• Closed unit ball: B, B d .

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 ⊂ ℓ 2 and t ∈ K, we consider the map The image is an N(0, t 2 ) variable, and the collection {X t ,t ∈ K} is called an isonormally-indexed Gaussian process in view of the isometric-isomorphism: Another key point is the identification

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: If X is bounded in expected norm, E X < ∞, then one has an expectation EX ∈ K , which can be given implicitly in terms of its support function There is a strong law of large numbers: Theorem 1 (Artstein and Vitale, 1975) If X 1 , X 2 , . . .are independent and identically distributed random convex bodies with E X 1 < ∞, then 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: 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 I R d is where the constants V j (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 L 1 , . . ., L n , such that EL 1 = B d .By the strong law of large numbers, as n → ∞, and so For one line segment (i.e., n = 1), one has where Π L ⊥ 1 signifies projection onto the subspace orthogonal to the one spanned by L 1 .By induction, where L S = ∑ i∈S L i .This can be re-expressed as where has the form of a U-statistic.Then where Π j signifies projection onto a random subspace of dimension j and Evol j (Π j K) .
(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, It is also true that where Z [ j,d] ,is a j × d matrix of independent N(0, 1) variables.This can be used (Vitale, 2008) to show Next we identify some of the intrinsic volumes:

EXTENSION OF INTRINSIC VOLUMES TO CONVEX BODIES IN ℓ 2
The extension of intrinsic volumes to convex bodies in Hilbert space and specifically to ℓ 2 was undertaken by Sudakov (1971) and Chevet (1976).To begin, let us identify the following collections: 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 ∈ K , define GB stands for "Gaussian Bounded" (Dudley, 1967) and refers to the following identification.
Some canonical cases are given in the next example.
Example Given a decreasing sequence of positive constants {a n } and an orthonormal set {e n , n = 1, 2, . ..}, set Example An example of an infinite-dimensional convex body that has no finite-dimensional analogue is as follows.Consider a map All Brownian motion bodies are the same, of course, up to an isometry.A particular realization in Theorem 4 (Gao and Vitale, 2001)

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 K n → K, but V j (K n ) → V j (K).In particular, one can have This leads to the following definition.
One has GC stands for "Gaussian Continuous" (Dudley, 1967), and the following gives the connection.

ITO-NISIO THEORY
Theorem 6 (Ito and Nisio, 1969) Suppose that t * ∈ K ∈ K GB .The oscillation of X at t * , given by The following elaborates this observation.
Theorem 7 (Vitale, 2001) Suppose that osc (t * ) > 0. Then (3) )) in the sense that for each j, the limit in Eq. 3 is equal to

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: