Some extensions of the class of convex bodies

TL;DR

This paper introduces a new class of epsilon-convex bodies extending convex bodies, explores their properties, connections to classical convex geometry results, and applications in geometric tomography, including stability theorems.

Contribution

It defines epsilon-convex bodies, studies their properties, and generalizes stability theorems, linking them to classical convex geometry and tomography.

Findings

Epsilon-convex bodies extend convex bodies in metric and normed spaces.

Connections established between epsilon-convex bodies and Helly's theorem.

Generalization of stability theorems by Groemer.

Abstract

We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, and find applications to geometric tomography. We introduce the notion of a circular projection and investigate the problem of determination of $\eps$-convex bodies by their projection-type images. The results generalize corresponding stability theorems by H. Groemer.