Reduced polygons in the hyperbolic plane (Q6614088)

The author defines the width, determined by a hyperplane, of a convex body in a hyperbolic space of given dimension as the distance between the given supporting hyperplane as the distance between the hyperplane and a most distant ultra-parallel hyperplane supporting the convex body. Also, the thickness of the convex body is defined as its infimum width over all supporting hyperplanes. If a the thickness of a convex body do not exceed the thickness of all the convex bodies properly contained in it then it is called a reduced body. A special class of reduced polygons is defined in this paper, analogous to the classes of all reduced polygons in the two dimensions Euclidean space and spherical space. It is called the ordinary reduced polygons. Properties concerning the width and thickness of ordinary reduced polygons is presented in this paper. The diameter of any ordinary reduced polygon in terms of its thickness is estimated. Some open questions on the diameter, perimeter, circumradius, and inradius of ordinary reduced polygons conclude the paper.