On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube

What the math gives to ML

The paper supplies a sharp homogeneous polynomial inequality for eight nonnegative variables indexed by the vertices of a 3-dimensional hypercube. Its right-hand side is the product of six coordinate-facet projection sums, while the left-hand side is a structured combination of triple and paired interactions; this gives a nontrivial measure of whether mass is distributed coherently across all three binary coordinates. A plausible neural-network transfer is to use the inequality as a hypercube-aware diversity score for eight-way MoE routing or channel grouping, rather than as a feasibility constraint, since the theorem already holds for every nonnegative vector. The resulting regularizer can favor routers whose probability mass has strong balanced projections across multiple binary partitions while avoiding arbitrary pairwise diversity penalties.