Width of convex bodies in hyperbolic space

Abstract

For every 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 $C$. We prove that if $\width_H (C) = \Delta (C)$ and if there exists a unique most distant point $j \in C$ from $H$, then the projection of $j$ onto $H$ belongs to $H \cap C$. We verify that the diameter of $C$ equals to the maximum width of $C$. We define bodies of constant width in $\mathbb{H}^d$ in the standard way as bodies whose all widths are equal. We show that every body of constant width is strictly convex. The minimum width of $C$ over all supporting $H$ is called the thickness $\Delta (C)$ of $C$. A convex body $R \subset \mathbb{H}^d$ is said to be reduced if $\Delta (Z) < \Delta (R)$ for every convex body $Z$ properly contained in $R$. We show that regular tetrahedra in $\mathbb{H}^3$ are not reduced. Similarly as in the Euclidean and spherical spaces, we introduce complete bodies and bodies of constant diameter in $\mathbb{H}^d$. We show that every body of constant width $\delta$ is a body of constant diameter $\delta$ and a complete body of diameter $\delta$. Moreover, the two last conditions are equivalent.