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.
