Large Shadows from Sparse Inequalities

TL;DR

This paper demonstrates that certain high-dimensional polytopes defined by sparse inequalities can have exponentially large shadows, revealing new geometric properties of these structures.

Contribution

It proves that the $d$-dimensional Klee-Minty cube also has an exponentially large shadow, extending known results from the Goldfarb cube.

Findings

Goldfarb cube's shadow contains all vertices

Klee-Minty cube's shadow also has $2^d$ vertices

Shadows of polytopes with one-variable inequalities are bounded by 2d

Abstract

The $d$-dimensional Goldfarb cube is a polytope with the property that all its $2^d$ vertices appear on some \emph{shadow} of it (projection onto a 2-dimensional plane). The Goldfarb cube is the solution set of a system of 2d linear inequalities with at most 3 variables per inequality. We show in this paper that the $d$-dimensional Klee-Minty cube --- constructed from inequalities with at most 2 variables per inequality --- also has a shadow with $2^d$ vertices. In contrast, with one variable per inequality, the size of the shadow is bounded by 2d.