  A long-standing sample compression conjecture asks to linearly bound the size
of the optimal sample compression schemes by the Vapnik-Chervonenkis (VC)
dimension of an arbitrary class. In this paper, we explore the rich metric and
combinatorial structure of oriented matroids (OMs) to construct proper
unlabeled sample compression schemes for the classes of topes of OMs bounded by
their VC-dimension. The result extends to the topes of affine OMs, as well as
to the topes of the complexes of OMs that possess a corner peeling. The main
tool that we use are the solutions of certain oriented matroid programs.
