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

For a hyperplane H supporting a convex body C in the hyperbolic space Hᵈ 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 Δ(C). A convex body R ⊂ Hᵈ is called reduced if for every body Z R we have Δ(Z) < Δ(R). We show that for any extreme point e of a reduced body R ⊂ Hᵈ 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 Hᵈ are defined as bodies whose all widths are equal. We prove that every complete body in Hᵈ is a body of constant width.