A Lost Counterexample and a Problem on Illuminated Polytopes

TL;DR

This paper links two historical geometric problems using Gale duality, revealing a counterexample to Marcus' conjecture and addressing an open question about illuminated polytopes' simpliciality.

Contribution

It uncovers a lost counterexample to Marcus' conjecture and answers an open problem about the structure of illuminated polytopes using Gale duality.

Findings

Mani's study provides a counterexample to Marcus' conjecture.

The counterexample has parameters exactly as Marcus claimed.

An answer is given to whether minimal illuminated polytopes can be nonsimplicial.

Abstract

In a Note added in proof to a 1984 paper, Daniel A. Marcus claimed to have a counterexample to his conjecture that a minimal positively k-spanning vector configuration in R^m has size at most 2km. However, the counterexample was never published, and seems to be lost. Independently, ten years earlier, Peter Mani in 1974 solved a problem by Hadwiger, disproving that every ``illuminated'' d-dimensional polytope must have at least 2d vertices. These two studies are related by Gale duality, an elementary linear algebra technique devised by Micha A. Perles in the sixties. Thus, we note that Mani's study provides a counterexample for Marcus' conjecture with exactly the parameters that Marcus had claimed. In the other direction, with Marcus' tools we provide an answer to a problem left open by Mani: Could ``illuminated'' d-dimensional polytopes on a minimal number of vertices be nonsimplicial?