On strong unimodality of multivariate discrete distributions

A discrete function f defined on Zn is said to be logconcave if f(λx+(1− λ)y) ≥ [f(x)]λ[f(y)]1−λ for x, y, λx + (1− λ)y ∈ Zn. A more restrictive notion is strong unimodality. Following Barndorff-Nielsen (1973) a discrete function p(z), z ∈ Zn is called strongly unimodal if there exists a convex function f(x), x ∈ Rn such that f(x) = − log p(x), if x ∈ Zn. In this paper sufficient conditions are given that a discrete function is strongly unimodal. Six sufficient conditions are given for the case of n = 3 and one for the general case. A three-dimensional example shows that the logconcavity of a discrete function does not imply strong unimodality, in general. Examples are presented.