An in-depth study of ball-bodies

TL;DR

This paper explores the properties and inequalities of ball-bodies, a class of convex bodies formed by intersections of Euclidean balls, linking them to key problems in convex geometry.

Contribution

It provides new insights into the structure, inequalities, and symmetrizations of ball-bodies, advancing understanding of bodies of constant width and related conjectures.

Findings

Analysis of isoperimetric and Santaló inequalities for ball-bodies

Characterization of boundary structure and curvature relations

Initial results on bodies of constant width

Abstract

In this paper we study the class of so called `ball-bodies' in ${\mathbb R}^n$, given by intersections of translates of Euclidean unit balls (or, equivalently, summand of the Euclidean ball). We study the class along with the natural duality operator defined on it. The class is naturally linked to many interesting problems in convex geometry, including bodies of constant width and the Knesser-Poulsen conjecture. We discuss old and new inequalities of isoperimetric type and of Santal\'{o} type, in this class. We study the boundary structure of bodies in the class, Carath\'eodory type theorem and curvature relations. We discuss various symmetrizations with relation to this class, and make some first steps regarding problems for bodies of constant width.