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A Steiner type formula for convex functions

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Andrea Colesanti
Andrea Colesanti
Affiliation:
Dipartimento di Matematica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy.
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Abstract

Given a convex function u, defined in an open bounded convex subset Ω of ℝn, we consider the set

where η is a Borel subset of Ω,ρ is nonnegative, and ∂u(x) denotes the subgradient (or subdifferential) of u at x. We prove that Pp(u; η) is a Borel set and its n-dimensional measure is a polynomial of degree n with respect to ρ. The coefficients of this polynomial are nonnegative measures defined on the Borel subsets of Ω. We find an upper bound for the values attained by these measures on the sublevel sets of u. Such a bound depends on the quermassintegrals of the sublevel set and on the Lipschitz constant of u. Finally we prove that one of these measures coincides with the Lebesgue measure of the image under the subgradient map of u.

MSC classification

Secondary: 52A22: Random convex sets and integral geometry
Type
Research Article
Information
Mathematika , Volume 44 , Issue 1 , June 1997 , pp. 195 - 214
Copyright
Copyright © University College London 1997

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