Convexity in (colored) affine semigroups

TL;DR

This paper extends classical convex geometry theorems to affine semigroups, introduces colored affine semigroups, and proves new theorems like Tverberg's and Helly's in this context, enriching the theory with applications such as a colored Frobenius number.

Contribution

It develops a new theory of colored affine semigroups and proves analogs of key convex geometry theorems within this framework, including Tverberg, Helly, and Caratheodory theorems.

Findings

Proved affine semigroup versions of Helly, Tverberg, and Caratheodory theorems.

Introduced colored affine semigroups and established their properties.

Developed a colored Frobenius number for numerical semigroups.

Abstract

In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Caratheodory. Additionally, we develop a new theory of colored affine semigroups, where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg's theorem and colorful Helly's theorem for semigroups, as well as a version of colorful Caratheodory's theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.