Geometrical properties of polyconvex polytopes

Motivated by existence and semicontinuity problems in multidimensional calculus of variations, a number of generalized convexity notions for functions has been introduced since the 50s of the last century. In its hierarchical order, the most important of these notions are polyconvexity, quasiconvexity and rank-one convexity. 01) However, the necessity to employ these semiconvexity notions as well for point sets in the space R was recognized much later. 02) As a consequence, the properties of semiconvex sets are still less thoroughly investigated than those of semiconvex functions. The present paper, which originated in the context of the study of multidimensional control problems of Dieudonné-Rashevsky type with nonconvex data, 03) makes a contribution to the geometry of polyconvex sets and, particularly, of polyconvex polytopes. For the description of these sets, we introduce the following notations.