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.
