Abstract
A measure P in a linear space X is said to be convex in any sense if the inequality
holds for a sufficiently wide class of sets A, B, and some function g. In this section, we are going to clarify, in what sense Gaussian distributions are convex. The notion of convexity it related to a remarkable isoperimetric theorem asserting that, among all sets of the same measure, it is the half-space that has the smallest “surface area”. We first consider the properties of a standard Gaussian distribution in ℝn, and then extend the results we obtain to arbitrary Radon Gaussian measures. As a corollary, the estimates of large deviations and a qualitative picture of the distribution of a convex functional will be derived.
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© 1995 Springer Science+Business Media Dordrecht
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Lifshits, M.A. (1995). Convexity and the Isoperimetric Property. In: Gaussian Random Functions. Mathematics and Its Applications, vol 322. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8474-6_11
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DOI: https://doi.org/10.1007/978-94-015-8474-6_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4528-7
Online ISBN: 978-94-015-8474-6
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