论文标题
在$ \ mathbb {r}^d $中的限制点集中的凸多图形
Convex polytopes in restricted point sets in $\mathbb{R}^d$
论文作者
论文摘要
对于有限点集$ p \ subset \ mathbb {r}^d $，用$ \ text {diam}（p）$表示最大的比率与$ p $中最小距离的比率。令$ c_ {d，α}（n）$为最大的整数$ c $，以使任何$ n $ - 点集$ p \ subset \ subset \ mathbb {r}^d $一般位置，满足$ \ text {diam}（diam}（p）<α\ sqrt [d] {d] {d] {n} $，包含一个$ c $ - 我们通过显示正常数$β=β=β（d，α）$和$γ=γ（d）$，将$ c_ {d，α}（n）$确定为$ n \ to \ infty $的渐近学确定为$ n \ to \ infty $ γn^{\ frac {d-1} {d+1}} $ for $α\ geq 2 $。
For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, α}(n)$ be the largest integer $c$ such that any $n$-point set $P \subset \mathbb{R}^d$ in general position, satisfying $\text{diam}(P) < α\sqrt[d]{n}$, contains an $c$-point convex independent subset. We determine the asymptotics of $c_{d, α}(n)$ as $n \to \infty$ by showing the existence of positive constants $β= β(d, α)$ and $γ= γ(d)$ such that $βn^{\frac{d-1}{d+1}} \le c_{d, α}(n) \le γn^{\frac{d-1}{d+1}}$ for $α\geq 2$.
