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DTSTART:20230101T000000
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DTSTART;TZID=Asia/Seoul:20240123T163000
DTEND;TZID=Asia/Seoul:20240123T173000
DTSTAMP:20260416T082839
CREATED:20231120T123044Z
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UID:7930-1706027400-1706031000@dimag.ibs.re.kr
SUMMARY:Zichao Dong\, Convex polytopes in non-elongated point sets in $\mathbb{R}^d$
DESCRIPTION: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.
URL:https://dimag.ibs.re.kr/event/2024-01-23/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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