An in-depth study of ball-bodies

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.