The isotropy constant and boundary properties of convex bodies

TL;DR

This paper demonstrates that convex bodies with positive generalized Gauss curvature at some boundary point cannot be local maximizers of the isotropy constant, revealing a geometric property linking curvature and isotropy.

Contribution

It establishes a new geometric criterion connecting boundary curvature with the extremal properties of the isotropy constant in convex bodies.

Findings

Convex bodies with positive curvature are not local maximizers of the isotropy constant.

The result links boundary curvature properties to isotropic extremal problems.

Provides insight into the geometric structure influencing the isotropy constant.

Abstract

Let ${\cal K}^n$ be the set of all convex bodies in $\mathbb R^n$ endowed with the Hausdorff distance. We prove that if $K\in {\cal K}^n$ has positive generalized Gauss curvature at some point of its boundary, then $K$ is not a local maximizer for the isotropy constant $L_K$.