Convex bodies of constant width in spheres
by
Lidiya Bazylevych
National University "Lviv Polytechnica"

The notion of convexity can be considered in arbitrary Riemannian manifold. We consider the standard Riemannian metric on the unit sphere S² of R³. Let U denote the open upper hemisphere od S². We say that a subset A of U is convex if every two points in A can be joined with a geodesic in A. We denote by cc(U) the hyperspace of compact convex subsets in U. One can easily prove that the hyperspace cc(U) is homeomorphic to the contractible Q-manifold Q\{*} (by Q we denote the Hilbert cube).

Note that a analogical result can be also obtained for the hyperspace of compact convex sets in the hyperbolic plane.

A convex set A in U is called a body of constant width d, where 0 < d < p/2, if, for every point a of the boundary of A, we have d(a, b) ≤ d and there exists a point c of the boundary of A such that d(a, c)=d.

Denote by cw_d(U) the set of bodies of constant width d in U. The main result states that this hyperspace is also homeomorphic to Q\{*}.

We leave as an open question whether an analogical result is valid for the hyperspace of compact convex bodies of constant width d in the hyperbolic plane H².

Date received: May 1, 2008