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 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 also 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.