A Monotone Unimodal Distribution which is Not Central Convex Unimodal

Abstract

For symmetric univariate distributions the usual definition of unimodality due to Khintchine has several equivalent formulations. When these concepts are generalized to higher dimensions in an attempt to define multivariate unimodality questions concerning their equivalence naturally arise. Of particular interest in this area is the relationship between two concepts first studied by Sherman and more recently by Dharmadhikari and Jogdeo. They asked if requiring that a distribution belong to the closed convex hull of all uniform distributions on symmetric convex bodies was the same as requiring that the probability it assigns to a symmetric convex set decrease as the set is translated away from the origin in a fixed direction. Sherman conjectured that the two concepts were the same while Dharmadhikari and Jogdeo felt that this was not so and they suggested a possible counterexample to Sherman's conjecture. In this paper it is shown that their example is indeed a counterexample.