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

In this paper, we explore properties of a family of probability density functions, called norm-induced densities, defined as 
where K is a n-dimensional convex set that contains the origin, 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. Since ft is log-concave only if p ≥ 1, this framework also covers nonlog-concave densities. Moreover, we establish a new set inclusion characterization for convex sets. This leads to a new concentration of measure phenomena for unbounded convex sets. Finally, these properties are used to develop an efficient probabilistic algorithm to test whether a convex set, represented only by membership oracles (a membership oracle for K and a membership oracle for its recession cone), is bounded or not, where the algorithm reports an associated certificate of boundedness or unboundedness.