Researcher ORCID Identifier
Open Access Senior Thesis
Bachelor of Science
© 2022 Amit Harlev
We introduce and prove Sperner’s lemma, the well known combinatorial analogue of the Brouwer fixed point theorem, and then attempt to gain a better understanding of the polytopal generalization of Sperner’s lemma conjectured in Atanassov (1996) and proven in De Loera et al. (2002). After explaining the polytopal generalization and providing examples, we present a new, simpler proof of a slightly weaker result that helps us better understand the result and why it is correct. Some ideas for how to generalize this proof to the complete result are discussed. In the last two chapters we provide a brief introduction to the basics of matroid theory before generalizing a matroid generalization of Sperner’s lemma proven in Lovász (1980) to polytopes. At the end we present some partial progress towards proving the polytopal generalization of Sperner’s lemma using this matroid generalization.
Harlev, Amit, "On the Polytopal Generalization of Sperner’s Lemma" (2022). HMC Senior Theses. 258.