EQUIDISTANT THICKNESS IN A SPACE OF CONSTANT CURVATURE

B. V. Dekster (Canada)

Abstract

First, we discuss here two existing (a la Santaló) definitions of minimum width of a convex body in an n-dimensional space Xn of constant curvature. These definitions are based on placing the body in a smallest part of Xn bounded by a pair of hyperplanes. They have some shortcomings. For instance, in the hyperbolic plane there exists a convex body of arbitrarily small minimum width that contains an arbitrary large circle. We come up with another definition of the minimum width called equidistant thickness of a compact set. It is based on placing the set in a smallest part of Xn between two hypersurfaces equidistant to a hyperplane. This preserves some natural properties of minimum width. Then we apply this new definition to estimate how thick at least a simplex in Xn should be when its edge lengths fall within a prescribed range.

Keywords and phrases: minimum width, spherical and hyperbolic geometries, equidistant thickness, simplex.