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- W4290802835 abstract "Let $ES_{d}(n)$ be the smallest integer such that any set of $ES_{d}(n)$ points in $mathbb{R}^{d}$ in general position contains $n$ points in convex position. In 1960, ErdH{o}s and Szekeres showed that $ES_{2}(n) geq 2^{n-2} + 1$ holds, and famously conjectured that their construction is optimal. This was nearly settled by Suk in 2017, who showed that $ES_{2}(n) leq 2^{n+o(n)}$. In this paper, we prove that $$ES_{d}(n) = 2^{o(n)}$$ holds for all $d geq 3$. In particular, this establishes that, in higher dimensions, substantially fewer points are needed in order to ensure the presence of a convex polytope on $n$ vertices, compared to how many are required in the plane." @default.
- W4290802835 created "2022-08-12" @default.
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- W4290802835 date "2022-08-09" @default.
- W4290802835 modified "2023-09-30" @default.
- W4290802835 title "Convex polytopes from fewer points" @default.
- W4290802835 doi "https://doi.org/10.48550/arxiv.2208.04878" @default.
- W4290802835 hasPublicationYear "2022" @default.
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