Ball-convex bodies and $L_p$ relative surface areas
Metric Geometry
2025-12-24 v1 Differential Geometry
Abstract
We define new surface area measures for ball-convex bodies which we call 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 relative surface areas.
Cite
@article{arxiv.2512.19836,
title = {Ball-convex bodies and $L_p$ relative surface areas},
author = {Elisabeth M. Werner and Diliya Yalikun},
journal= {arXiv preprint arXiv:2512.19836},
year = {2025}
}