Norm-Induced Densities and Testing the Boundedness of a Convex Set

Abstract

In this paper we explore properties of a family of probability density functions, called norm-induced densities, defined aswhere K is a n-dimensional convex set, parameters t > 0 and p > 0, and · is any norm. We also develop connections between these densities and geometric properties of K such as diameter, width of the recession cone, and others.Moreover, we establish a new set inclusion characterization for convex sets. This leads to a new concentration phenomena for unbounded convex sets. More explicitly, we show that most points of a convex set are contained in any enlargement of the set's recession cone.Finally, these properties are used to develop an efficient probabilistic algorithm to test whether a convex set, represented only by a membership oracle, is bounded or not, where the algorithm reports an associated certificate of boundedness or unboundedness.