The Minimal Perimeter of a Log-Concave Function

Niufa Fang,Zengle Zhang

doi:10.3390/math8081365

Abstract

Inspired by the equivalence between isoperimetric inequality and Sobolev inequality, we provide a new connection between geometry and analysis. We define the minimal perimeter of a log-concave function and establish a characteristic theorem of this extremal problem for log-concave functions analogous to convex bodies.

Highlights

The isoperimetric inequality is an important inequality in geometry which originated from the well-known isoperimetric problem
Motivated by the work of Giannopoulos and Milman [7], we consider the extremal problems of log-concave functions instead of convex bodies, and our purpose is to discuss the possibility of an isometric approach to these questions
Many outstanding works showed that the log-concave function is closely linked to the convex body

Summary

Introduction

The isoperimetric inequality is an important inequality in geometry which originated from the well-known isoperimetric problem. Our purpose is to study the isotropic position of log-concave functions. K x k2 is the Gaussian function and k x k is the Euclidean norm of x ∈ Rn. Motivated by the work of Giannopoulos and Milman [7], we consider the extremal problems of log-concave functions instead of convex bodies, and our purpose is to discuss the possibility of an isometric approach to these questions. Theorem 1 implies that the log-concave function f has minimal perimeter if and only if μ f (·) is isotropic, and provides a further example of the connections between the theory of convex bodies and that of functions. We remark that our works belong to the asymptotic theory of log-concave functions which parallel to that of convex bodies. We hope that our work provides some useful tools or ideas in the development of geometry of log-concave functions

Preliminaries

Minimal Perimeter of Log-Concave Functions

Conclusions