Mathematics Colloquia and Seminars

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For a closed, convex surface $S$ in Euclidean space, the Far-Point map takes $x\in S$ to the set of points $F_x$ of maximal intrinsic distance from $x$. The past 15 years have seen significant advancements regarding for which surfaces the Far-Foint map is single-valued, involutive, injective, and possessing an everywhere-defined inverse. In this talk I take the special case of the regular octahedron, and give an explicit description of the Far-Point map on the octahedron. This provides insight into how the Far-Point map can fail to be single-valued and involutive, while being injective everywhere but on a set of measure zero. Interestingly, we show that the Far-Point map on the octahedron is piece-wise birational and is almost-projective is a way that links it to the cremona group.