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A Steiner type formula for convex functions
Author(s) -
Colesanti Andrea
Publication year - 1997
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300012067
Subject(s) - mathematics , subderivative , subgradient method , lebesgue measure , combinatorics , measure (data warehouse) , lipschitz continuity , bounded function , borel measure , type (biology) , function (biology) , discrete mathematics , convex set , polynomial , lebesgue integration , regular polygon , probability measure , mathematical analysis , convex optimization , mathematical optimization , geometry , ecology , database , evolutionary biology , computer science , biology
Given a convex function u , defined in an open bounded convex subset Ω of ℝ n , we consider the setP ρ ( u ; η ) = { x + ρ v : x ∈ η , v ∈ ∂ u ( x ) } ,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 P p ( 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 .