Some extensions of the class of convex bodies
Vladimir Golubyatnikov, Vladimir Rovenski

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 -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 -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 -convex bodies by their projection-type images. The results generalize corresponding stability theorems by H. Groemer.
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