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$4m-3$ Lattice Points in the Plane

Student-Run Applied & Math Seminar

Speaker: Thomas Hogan, UC Davis
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Location: 3106 MSB
Start time: Fri, Apr 20 2018, 12:10PM

Tverberg's theorem says that sufficiently many points in euclidean space can always be partitioned into $m$ subsets so that the intersection of the convex hulls of the $m$ subsets is non-empty. In the 1970's Doignon proposed a variant of Tverberg's theorem where the points are required to have integer coordinates. We will focus on this variant throughout the talk, and sketch the proof that any $4m−3$ (for $m≥3$) lattice points in the plane can be colored $m$ colors so that there is a lattice point in the intersection of the convex hulls of the $m$ colors. We will also discuss the analogous problem in higher dimensions.

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