Travis Dillon

collaborators

8 papers

math.MG2026

Small lattice polytopes have few vertices

Travis Dillon

This is a geometric retelling of Konyagin and Sevast'yanov's proof of Andrew's theorem, which is a tight upper bound on the number of vertices of a d-dimensional lattice polytope i…

math.ST2026

Fixed-strength spherical designs

Travis Dillon

A spherical $t$ -design is a finite subset $X$ of the unit sphere such that every polynomial of degree at most $t$ has the same average over $X$ as it does over the entire sphere. D…

math.CO2025

At most 10 cylinders mutually touch: a Ramsey-theoretic approach

Travis Dillon, Junnosuke Koizumi, Sammy Luo

Littlewood asked for the maximum number $N$ of congruent infinite cylinders that can be arranged in $R^{3}$ so that every pair touches. We improve upon the proof of the seco…

math.MG2025

Closed curve covering and multiagent TSP ratios

Travis Dillon, Adrian Dumitrescu

How efficiently can a closed curve of unit length in $R^{d}$ be covered by $k$ closed curves so as to minimize the maximum length of the $k$ curves? We show that the maximum…

math.MG2025

Quantitative selection theorems

Travis Dillon

The point selection theorem says that the convex hull of any finite point set contains a point that lies in a positive proportion of the simplices determined by that set. This pape…

math.MG2025

Hollow polytopes with many vertices

Srinivas Arun, Travis Dillon

Given a set $S \subseteq R^{d}$ , a hollow polytope has vertices in $S$ but contains no other point of $S$ in its interior. We prove upper and lower bounds on the maximum num…