Illumination bodies for ball-convex bodies

Brandon Oliva; Elisabeth M. Werner; Diliya Yalikun

Illumination bodies for ball-convex bodies

Metric Geometry 2026-07-15 v1 Differential Geometry

Authors: Brandon Oliva , Elisabeth M. Werner , Diliya Yalikun

Abstract

We introduce illumination bodies and weighted illumination bodies in the class of $n$ -dimensional $R$ -ball convex bodies. These bodies may be regarded as dual counterparts of the recently introduced $R$ -ball floating bodies. We prove that the illumination bodies are convex. We also show that a left derivative of volume gives rise to a surface area measure for ball-convex bodies, the relative surface area measure. In dimension $2$ , we establish an isoperimetric inequality for this relative surface area.

Keywords

illumination bodies ball-convex bodies floating bodies surface area measure isoperimetric inequality

Cite

@article{arxiv.2607.13714,
  title  = {Illumination bodies for ball-convex bodies},
  author = {Brandon Oliva and Elisabeth M. Werner and Diliya Yalikun},
  journal= {arXiv preprint arXiv:2607.13714},
  year   = {2026}
}