Hidden Vertices in Extensions of Polytopes

TL;DR

This paper investigates the properties of extension polytopes for polygons and higher-dimensional polytopes, revealing that minimal extensions often have vertices that do not correspond to the original polytope's vertices.

Contribution

It proves that for certain polygons and polytopes, minimal extensions do not necessarily project onto the original vertices, challenging previous assumptions about extension properties.

Findings

Minimum size extensions of heptagons in general position lack vertex projection property.

At least 1/9 of vertices in minimal extensions of certain d-polytopes are not projections of original vertices.

Results apply to a broad class of polytopes, impacting extension theory.

Abstract

Some widely known compact extended formulations have the property that each vertex of the corresponding extension polytope is projected onto a vertex of the target polytope. In this paper, we prove that for heptagons with vertices in general position none of the minimum size extensions has this property. Additionally, for any d >= 2 we construct a family of d-polytopes such that at least 1/9 of all vertices of any of their minimum size extensions is not projected onto vertices.