Travis Dillon

collaborators

6 papers

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

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

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

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…

math.ST2025

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.CO2024

The prime grid contains arbitrarily large empty polygons

Travis Dillon

This paper proves a 2017 conjecture of De Loera, La Haye, Oliveros, and Roldán-Pensado that the "prime grid" $\big\{(p,q) \in \mathbb{Z}^2 : \text{$p $an d$ q$ are prime}\big\} \sub…