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Semi-Interlaced Polytopes
Moscow Mathematical Journal. 2026. Vol. 26. No. 2. P. 167–187.
Селянин Ф. И.
Minkowski mixed volume of n subpolytopes D1,…,Dn of a polytope P⊂Rn clearly does not exceed the normalized volume n!Vol(P). Equality holds if and only if the subpolytopes are interlaced, i.e., each proper face F⊊P intersects at least dim(F)+1 of the polytopes Di. Efficiently computing mixed volumes for more general collections of subpolytopes is crucial for estimating the complexity of numerically solving polynomial systems.
Motivated by relaxing the bound dim(F)+1 to dim(F), we prove a combinatorial formula for the mixed volume of a broad class of semi-interlaced polytopes. This class includes, in particular, the off-coordinate polytopes used in computing algebraic degrees—such as Maximum Likelihood, Euclidean Distance, and Polar degrees—via the Kouchnirenko–Bernshtein theory. We also present applications of our results to the Arnold monotonicity problem (Problem 1982-16), which concerns the dependence of Milnor numbers on the Newton polyhedra
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