Graduation Year
2020
Date of Submission
12-2019
Document Type
Open Access Senior Thesis
Award
Best Senior Thesis in Mathematics
Degree Name
Bachelor of Arts
Department
Mathematics
Reader 1
Lenny Fukshansky
Terms of Use & License Information
Abstract
This thesis explores several problems in discrete geometry, focusing on covering problems. We first go over some well known results, explaining Keith Ball's solution to the symmetric Tarski plank problem, as well as results of Alon and F\"uredi on covering all but vertices of a cube with hyperplanes. The former extensively utilizes techniques from matrix analysis, and the latter applies polynomial method. We state and explore the related problem, asking for the number of parallel hyperplanes required to cover a given discrete set of points in $\mathbb{Z}^{d}$ whose entries are bounded, and prove that there exist sets which are ``difficult'' to cover in every dimension for entries whose absolute values are bounded by~1 using a similar polynomial-based approach.
Recommended Citation
Hsu, Alexander, "Discrete Geometry and Covering Problems" (2020). CMC Senior Theses. 2289.
https://scholarship.claremont.edu/cmc_theses/2289
