Facially exposed cones are not always nice

TL;DR

This paper investigates Gabor Pataki's conjecture that all facially exposed cones are nice, confirming it in three dimensions but providing a counterexample in four dimensions.

Contribution

It proves the conjecture holds in three dimensions and presents a counterexample in four dimensions, showing the conjecture does not hold universally.

Findings

Conjecture is true in 3D cases.

Counterexample exists in 4D.

Facially exposed cones are not always nice.

Abstract

We address the conjecture proposed by Gabor Pataki that every facially exposed cone is nice. We show that the conjecture is true in the three-dimensional case, however, there exists a four-dimensional counterexample of a cone that is facially exposed but is not nice.