Travis Dillon

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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…