Complete Equality and Sharp Minimality for Marked Two-Probe Mixed-Volume Deficits of Zonoids

Abstract

We study a mixed-volume deficit for zonoids determined by a base family and two marked probes in dimension d. The deficit Δ = βγ − αη has a formal Gale representation. On its rank-two layer, k = 2 (equivalently g = d), the deficit is nonnegative and its equality set is explicit. Combining this formal result with a necessary-and-sufficient realizability criterion and a rank-stratified analysis of the primal configurations yields the equality set across all actual ranks. The same pointwise geometry gives the equality conditions for finite atomic and general generating measures, and the prescribed role-preserving symmetries and activity conditions can be verified on the successful stratum. For every d ≥ 2, the resulting closure-and-activity criterion is first attained at base complexity g = d. In dimension one, the identity Δ₁,₍g₎ = |b||c| rules out the required positive non-atomic zero witness for every g, so no finite successful base size exists. Consequently, g_min(d,2) = +∞ for d = 1 and g_min(d,2) = d for d ≥ 2; on the finite branch, N_min = d + 2 and k_min = 2. Thus k = 2 is the minimal successful layer for the stated closure-and-activity criterion; it is not asserted to be the maximal universal positivity range.