Covariance bounds for multivariate unimodal distributions and a characterization of uniformity

A univariate probability distribution which has support in [-1, 1] and is unimodal with respect to 0 has a variance of at most , which is attained only for the uniform distribution. This well-known result is extended to the multivariate case, where the variance is replaced by the generalized or mean variance and a new definition of unimodality is used.