Further results on increasing paths in the one-skeleton of a convex body

Abstract

Various problems are considered in an attempt to generalize the simplex algorithm of linear programming to a much wider class of convex bodies than the class of convex polytopes. A conjecture of D.G. Larman and C.A. Rogers is disproved by constructing a three-dimensional convex body K with an extreme point e, so that for a certain linear functional f, there are no paths in the one-skeleton of K leading from e, along which f strictly increases. Their conjectured generalization is, however, proved for the large class of three-dimensional convex bodies, all of whose extreme points are exposed. A strong generalization of the simplex algorithm is obtained for the class of all finite-dimensional convex bodies, where, for a given exposed point e of a convex body K, it is possible to find f-strictly-increasing paths in the one-skeleton of K, leading from e, for almost all linear functionals f.