Illuminating spindle convex bodies and minimizing the volume of spherical sets of constant width

TL;DR

This paper extends Schramm's theorem to 'fat' spindle convex bodies, providing insights into their illumination and volume minimization within the class of convex bodies of constant width.

Contribution

It generalizes existing results on convex bodies of constant width to include 'fat' spindle convex bodies, expanding understanding of their geometric properties.

Findings

Extended Schramm's theorem to 'fat' spindle convex bodies.

Provided bounds and properties for illuminating these bodies.

Enhanced understanding of volume minimization in convex geometry.

Abstract

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a "fat" one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm's theorem and its proof on illuminating convex bodies of constant width to the family of "fat" spindle convex bodies.