## Gaussian processes and mixed volumes.(English)Zbl 0636.60036

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.)

Zbl 0188.205
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