The log-Minkowski inequality of curvature entropy for non-symmetric convex bodies

TL;DR

This paper extends the log-Minkowski inequality of curvature entropy from symmetric to general convex bodies in the plane, establishing key equivalences related to cone-volume measures and volume inequalities.

Contribution

It proves the log-Minkowski inequality of curvature entropy for non-symmetric convex bodies in the plane, generalizing previous symmetric case results.

Findings

Proves the plane log-Minkowski inequality of curvature entropy for general convex bodies.

Shows the equivalence between cone-volume measure uniqueness and log-Minkowski inequalities.

Establishes connections between volume and curvature entropy inequalities.

Abstract

In an earlier paper \cite{mazeng} the authors introduced the notion of curvature entropy, and proved the plane log-Minkowski inequality of curvature entropy under the symmetry assumption. In this paper we demonstrate the plane log-Minkowski inequality of curvature entropy for general convex bodies. The equivalence of the uniqueness of cone-volume measure, the log-Minkowski inequality of volume, and the log-Minkowski inequality of curvature entropy for general convex bodies in $\mathbb R^{2}$ are shown.