Theorems of Carath\'{e}odory, Minkowski-Weyl, and Gordan up to symmetry

TL;DR

This paper extends key classical theorems in polyhedral geometry to infinite-dimensional spaces with symmetric group invariance, broadening their applicability in advanced mathematical contexts.

Contribution

It introduces generalized versions of Carathéodory's theorem, Minkowski-Weyl theorem, and Gordan's lemma for symmetric-invariant cones and monoids in infinite-dimensional spaces.

Findings

Generalization of Carathéodory's theorem to infinite dimensions with symmetry.

Extension of Minkowski-Weyl theorem to symmetric cones in infinite-dimensional spaces.

Adaptation of Gordan's lemma for symmetric monoids in infinite-dimensional settings.

Abstract

In this paper we extend three classical and fundamental results in polyhedral geometry, namely, Carath\'{e}odory's theorem, the Minkowski-Weyl theorem, and Gordan's lemma to infinite dimensional spaces, in which considered cones and monoids are invariant under actions of symmetric groups.