Minkowski-Weyl theorem and Gordan's lemma up to symmetry

TL;DR

This paper extends classical convex geometry theorems to infinite-dimensional symmetric settings, providing new local-global principles, classifications, and proofs for equivariant cones and monoids under the infinite symmetric group.

Contribution

It introduces equivariant versions of the Minkowski--Weyl theorem and Gordan's lemma in an infinite-dimensional context, with classifications and stabilization results.

Findings

Established a local equivariant Minkowski--Weyl theorem

Proved local-global principles for equivariant finite generation

Classified non-pointed symmetric cones and non-positive symmetric normal monoids

Abstract

We investigate equivariant analogues of the Minkowski--Weyl theorem and Gordan's lemma in an infinite-dimensional setting, where cones and monoids are invariant under the action of the infinite symmetric group. Building upon the framework developed earlier, we extend the theory beyond the nonnegative case. Our main contributions include a local equivariant Minkowski--Weyl theorem, local-global principles for equivariant finite generation and stabilization of symmetric cones, and a full proof of the equivariant Gordan's lemma. We also classify non-pointed symmetric cones and non-positive symmetric normal monoids, addressing new challenges in the general setting.