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