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 f t(x)={e ^-t||x||p/∫ Ke ^-t||y||pdy, x ∈ K 0, x ∉ K, 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 f t 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. © 2008 INFORMS.

Duke Scholars

Author Alexandre Belloni Fuqua School of Business