Lower bound results for conditionally decomposable polytopes

TL;DR

This paper investigates the minimal vertex counts of conditionally decomposable polytopes, establishing bounds and exact values in various dimensions, and explores implications for the face counts of decomposable polytopes.

Contribution

It provides new lower bounds and exact minimal counts for vertices and facets of conditionally decomposable polytopes across different dimensions.

Findings

Minimum vertices of conditionally decomposable d-polytopes are in [3d-3, 4d-4].

For polytopes with a line segment summand, 4d-4 vertices are sharp.

Minimum facets of conditionally decomposable polytopes are 9 in 4D and d+4 in higher dimensions.

Abstract

It is possible for a combinatorial type of polytope to have both decomposable and indecomposable realizations; here decomposability is meant with respect to Minkowski addition. Such polytopes are called conditionally decomposable. We show that the minimum number of vertices of a conditionally decomposable $d$-polytope is in the range $[3d-3, 4d-4]$, and that for a polytope having a line segment for a summand, $4d-4$ is sharp. As an application, the exact lower bound of the number of $k$-faces of a decomposable $d$-polytope with $2d+m$ vertices ($2 \le m\le d-4$) is obtained. Concerning the facets, in dimension 4, the minimum number of facets of a conditionally decomposable polytope is 9, and in dimension $d\ge 5$, the minimum is $d+4$.