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Zichao Dong, Convex polytopes in non-elongated point sets in $\mathbb{R}^d$

Tuesday, January 23, 2024 @ 4:30 PM - 5:30 PM KST

Room B332, IBS (기초과학연구원)


Zichao Dong
IBS Extremal Combinatorics and Probability Group

For any finite point set $P \subset \mathbb{R}^d$, we denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, \alpha}(n)$ be the largest integer $c$ such that any $n$-point set $P \subset \mathbb{R}^d$ in general position, satisfying $\text{diam}(P) < \alpha\sqrt[d]{n}$ (informally speaking, `non-elongated’), contains a convex $c$-polytope. Valtr proved that $c_{2, \alpha}(n) \approx \sqrt[3]{n}$, which is asymptotically tight in the plane. We generalize the results by establishing $c_{d, \alpha}(n) \approx n^{\frac{d-1}{d+1}}$. Along the way we generalize the definitions and analysis of convex cups and caps to higher dimensions, which may be of independent interest. Joint work with Boris Bukh.


Tuesday, January 23, 2024
4:30 PM - 5:30 PM KST
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Room B332
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