Convex hulls in the hyperbolic space |
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Authors: | Itai Benjamini Ronen Eldan |
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Institution: | 1. Department of Mathematics, Weizmann Institute, Rehovot, Israel 2. School of Mathematical Sciences, Tel-Aviv University, 69978, Tel-Aviv, Israel
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Abstract: | We show that the convex hull of any N points in the hyperbolic space ${\mathbb{H}^{n}}$ is of volume smaller than ${\frac{2 (2 \sqrt \pi)^n}{\Gamma(\frac n 2)} N}$ , and that for any dimension n there exists a constant C n > 0 such that for any set ${A \subset \mathbb{H}^{n}}$ , $$Vol(Conv(A_1)) \leq C_n Vol(A_1)$$ where A 1 is the set of points of hyperbolic distance to A smaller than 1. |
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