Gaussian processes and mixed volumes. (English) Zbl 0636.60036

Milman, V. D.; Pisier, G.

Ann. Probab. 15, 292-304 (1987).

Let K be a compact subset of a real Hilbert space H and denote \(V_ n=\sup \{vol(P(K))\}\) where the supremum runs over all orthogonal projections \(P: H\to H\) of rank n and vol denotes n-dimensional volume. Let \[ E V(K)=\lim_{n\to \infty}\sup (\log V_ n(K)/(n \log n)) \] be the exponent of the volume of K. Let \(X_ t\) be any Gaussian process indexed by H such that \(\| X_ t-X_ s\|_ 2=\| t-s\|\). A set K is called G.C. (resp. G.B.) if the process \(X_ t\) has a version with continuous (resp. bounded) paths on K. R. M. Dudley [J. Funct. Anal. 1, 290-330 (1967; Zbl 0188.205)] showed that E V(K)\(\leq -1\) for any G.B. set. This paper shows that if E V(K)\(<-1\) then K is a G.C. set.

Reviewer: J.Cuzick

MSC:

60G15	Gaussian processes
60G17	Sample path properties
41A46	Approximation by arbitrary nonlinear expressions; widths and entropy
47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)

Keywords:

continuity of Gaussian processes; metric entropy; exponent of volume; compact subset of a real Hilbert space

Citations:

Zbl 0188.205