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Jesús A. De Loera
deloera@math.ucdavis.edu
Jesús A. De Loera
Mathematics Department, University of California Davis, Davis, CA 95616, USA
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Christopher O’Neill
cdoneill@sdsu.edu
Christopher O’Neill
Mathematics Department, San Diego State University, San Diego, CA 92182, USA
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Chengyang Wang
cyywang@ucdavis.edu
Chengyang Wang
Mathematics Department, University of California Davis, Davis, CA 95616, USA
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In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Carathéodory. 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 Carathéodory’s theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number.
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