The Minimal Perimeter of a Log-Concave Function

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.

Keywords

• isoperimetric problem
• minimal perimeter
• log-concave functions
• isotropic measure