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A Steiner type formula for convex functions
Published online by Cambridge University Press: 26 February 2010
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.
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- Research Article
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- Copyright © University College London 1997
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