Applications of equidistant supporting surfaces of a convex body in the hyperbolic space

Abstract

For a hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. The thickness (i.e., the minimum width) of $C$ is denoted by $\Delta(C)$. A convex body $R \subset \mathbb{H}^d$ is called reduced if for every body $Z \subsetneq R$ we have $\Delta(Z) < \Delta(R)$. We show that for any extreme point $e$ of a reduced body $R \subset \mathbb{H}^d$ there exists a supporting hyperplane $H$ of $R$ which passes through $e$ or its equidistant surface supporting $R$ passes through $e$. Bodies of constant width in $\mathbb{H}^d$ are defined as bodies whose all widths are equal. We prove that every complete body in $\mathbb{H}^d$ is a body of constant width.