Some extensions of the class of $k$-convex bodies

TL;DR

This paper extends the theory of $k$-convex and $k$-visible bodies in Euclidean spaces by introducing circular projections and related classes, generalizing classical geometric tomography results and exploring new applications.

Contribution

It introduces circular projections in normed spaces and studies $k$-circular convex and visible bodies, broadening the scope of geometric tomography.

Findings

Generalized classical results of geometric tomography.

Developed new classes of bodies related to circular projections.

Established applications of these generalized bodies.

Abstract

We study relations of some classes of $k$-convex, $k$-visible bodies in Euclidean spaces. We introduce and study \textrm{circular projections} in normed linear spaces and classes of bodies related with families of such maps, in particular, \textrm{$k$-circular convex} and \textrm{$k$-circular visible} ones. Investigation of these bodies more general than $k$-convex and $k$-visible ones allows us to generalize some classical results of geometric tomography and find their new applications.