Dimension thresholds for log-concavity of normalized mixed moments

Abstract

For an arbitrary nonnegative coordinatewise nondecreasing integrable weight of positive integral on a finite d-dimensional box, we prove joint log-concavity of its mixed moments divided by ∏ᵢ Γ(αᵢ + 1) on αᵢ ≥ d/4, d ≥ 2. Finite product grids with ordered log-concave coordinate values and increasing index weights admit the sufficient orthant αᵢ ≥ d/4 + 4. The weights may have dependent coordinates and need only be monotone. A signed-block covariance argument gives larger convex anisotropic domains, allowing large exponents in some coordinates to compensate for smaller ones in others. For example, in dimension 100 it admits continuous exponents (1,100,…,100) and grid exponents (6,100,…,100). Increasing corner weights give counterexamples at common exponent d/4 − O(1), so the universal leading dimension coefficient 1/4 is optimal in both classes. Conditional logarithmic-deficit bounds and elementary trigamma inequalities supply the estimates. The same covariance mechanism yields joint spatial log-concavity for product-power smoothing of arbitrary decreasing densities and arrays, with coordinate-specific power conditions and the same optimal leading common-power coefficient. Exact rational witnesses make the obstructions reproducible. An attributed classical matroid consequence gives a dimension-free binary endpoint, contrasting with the dimension-linear powers required for arbitrary unweighted hereditary binary supports.