Ball-convex bodies and $L_p$ relative surface areas

TL;DR

This paper introduces new $L_p$ relative surface area measures for ball-convex bodies, demonstrating their invariance, establishing inequalities, and linking them to a novel entropy concept and geometric interpretations.

Contribution

It defines $L_p$ relative surface areas for ball-convex bodies and explores their properties, inequalities, and connections to entropy and floating bodies.

Findings

$L_p$ relative surface areas are rigid motion invariant valuations.

Established inequalities and monotonicity properties for these measures.

Provided a geometric interpretation via derivatives of volume with weighted floating bodies.

Abstract

We define new surface area measures for ball-convex bodies which we call $L_p$ relative surface areas. We show that those are rigid motion invariant valuations. We establish inequalities for these quantities and prove a monotonicity behavior which leads to a new notion of entropy for ball-convex bodies. We introduce a weighted ball floating body. A derivative of volume of a ball-convex body with a weighted ball floating body provides a geometric interpretation of the $L_p$ relative surface areas.