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

TL;DR

This paper derives exact formulas for the maximum number of faces of various dimensions in the Minkowski sum of three convex polytopes in any dimension, generalizing known results and identifying conditions for tight bounds.

Contribution

It provides the first tight expressions for face counts of Minkowski sums of three convex polytopes in arbitrary dimensions, extending previous two- and three-dimensional results.

Findings

Exact formulas for maximum face counts in Minkowski sums

Tight bounds achieved by specific polytopes on moment-like curves

Generalization of known 2D and 3D results to higher dimensions

Abstract

We derive tight expressions for the maximum number of $k$-faces, $0\le k\le d-1$, of the Minkowski sum, $P_1+P_2+P_3$, of three $d$-dimensional convex polytopes $P_1$, $P_2$ and $P_3$, as a function of the number of vertices of the polytopes, for any $d\ge 2$. Expressing the Minkowski sum of the three polytopes as a section of their Cayley polytope $\mathcal{C}$, the problem of counting the number of $k$-faces of $P_1+P_2+P_3$, reduces to counting the number of $(k+2)$-faces of the subset of $\mathcal{C}$ comprising of the faces that contain at least one vertex from each $P_i$. In two dimensions our expressions reduce to known results, while in three dimensions, the tightness of our bounds follows by exploiting known tight bounds for the number of faces of $r$ $d$-polytopes, where $r\ge d$. For $d\ge 4$, the maximum values are attained when $P_1$, $P_2$ and $P_3$ are $d$-polytopes, whose vertex sets are chosen appropriately from three distinct $d$-dimensional moment-like curves.