The maximum number of faces of the Minkowski sum of two convex polytopes

TL;DR

This paper provides exact formulas for the maximum number of faces of the Minkowski sum of two convex polytopes, identifying specific polytope configurations that achieve these maxima in different dimensions.

Contribution

It derives tight bounds for face counts of Minkowski sums and characterizes the extremal polytopes that attain these bounds based on dimension parity.

Findings

Maximum face counts are achieved by cyclic polytopes in even dimensions.

Maximum face counts are achieved by neighborly polytopes in odd dimensions.

Explicit formulas relate face counts to the number of vertices of the summand polytopes.

Abstract

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1\oplus{}P_2$, of two $d$-dimensional convex polytopes $P_1$ and $P_2$, as a function of the number of vertices of the polytopes. For even dimensions $d\ge{}2$, the maximum values are attained when $P_1$ and $P_2$ are cyclic $d$-polytopes with disjoint vertex sets. For odd dimensions $d\ge{}3$, the maximum values are attained when $P_1$ and $P_2$ are $\lfloor\frac{d}{2}\rfloor$-neighborly $d$-polytopes, whose vertex sets are chosen appropriately from two distinct $d$-dimensional moment-like curves.