Reduced polygons in the hyperbolic plane

For a hyperplane H supporting a convex body C in the hyperbolic space Hᵈ 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 minimum width of C over all supporting H is called the thickness Δ (C) Δ ( C ) of C . A convex body R ⊂ Hᵈ R ⊂ H d is said to be reduced if Δ (Z) < Δ (R) Δ ( Z ) < Δ ( R ) for every convex body Z properly contained in R . We describe a class of reduced polygons in H² H 2 and present some properties of them. In particular, we estimate their diameters in terms of their thicknesses.