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Abstract: |
Let K be a convex compact subset of a separable Hilbert space H. One of the most important geometric characteristics of K is its intrinsic volumes. In the finitedimensionalcase (K ? R^d), the intrinsic volumes V_k(K), k = 0, 1, . . . , d, are defined as the coefficients in the Steiner formula. Steiner proved that the volume of the λ-neighborhood of K is represented by a polynomial in λ with coefficients V_k(K) (where the normalization is chosen in a special way). It can be shown that the intrinsic volumes of the set do not depend on the dimension d of the ambient space Rd. This observation allowed Sudakov and Chevet to generalize the concept of intrinsic volume to the case of infinite-dimensional convex sets.
Later, Sudakov [1] and Tsirelson [2] discovered a deep connection between the intrinsic volumes of some convex compact sets and Gaussian processes on these sets. A generalization of intrinsic volumes are the so-called mixed volumes, which are defined in a similar way, namely, as the coefficients in the Minkowski's formula for the volume of a Minkowski sum of an arbitrary number of finite-dimensional compact sets. In this work, we generalize Tsirelson's theorem [2] to the mixed volumes of the infinite-dimensional convex compact subsets ofH, first introducing this notion.
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Press. - 2014. |
