On Mixed Quermassintegral for Log-Concave Functions

Fangwei Chen,Miao Luo,Congli Yang,Jianbo Fang,Youjiang Lin

doi:10.1155/2020/8811566

Fangwei Chen, Miao Luo + Show 3 more

Open Access

https://doi.org/10.1155/2020/8811566

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Abstract

In this paper, the functional Quermassintegral of log-concave functions in ℝ n is discussed. We obtain the integral expression of the i th functional mixed Quermassintegral, which is similar to the integral expression of the i th mixed Quermassintegral of convex bodies.

Highlights

Let Kn be the set of convex bodies in Rn, the fundamental BrunnMinkowski inequality for convex bodies states that for K, L ∈ Kn, the volume of the bodies and of their Minkowski sum K + L = fx + y : x ∈ K, y ∈ Lg is given by
The main result in this paper is to show that the ith functional mixed Quermassintegral has the following integral expressions
With equality if and only if K and L are homothetic; namely, they agree up to a translation and a dilation. Another geometric quantity related to the convex bodies K and L is the mixed volume

Summary

Introduction

We define the first variation of Wi at f along g, which is It is a natural extension of the Quermassintegral of convex bodies in Rn; we call it the ith functional mixed Quermassintegral. The main result in this paper is to show that the ith functional mixed Quermassintegral has the following integral expressions. Let f , g ∈ A ′, be integrable functions, μiðf Þ be the i-dimensional measure of f , and Wiðf , gÞ be the ith functional mixed Quermassintegral of f and g. Owing to the Blaschke-Petkantschin formula and the similar definition of the support function of f , we obtain the integral representation of the ith functional mixed Quermassintegral Wið f , gÞ

Preliminaries

Projection of Functions onto Linear Subspace

Findings

Functional Quermassintegrals of LogConcave Function