论文标题
直径的helly型定理
A mélange of diameter Helly-type theorems
论文作者
论文摘要
直径的Helly-type定理提供了一个有限的凸套装的直径，以$ \ MATHBB {r}^d $在所有足够小的亚家庭相交的直径上进行了一些信息。我们证明了Bárány，Katchalski和Pach的长期猜想的分数和丰富多彩的版本。我们还表明，Minkowski Norm在且仅当其单位球是多层室并证明具有五颜六色的版本的情况下，就可以直径确切的直径定理。最后，我们证明了``包含$ k $ colinear integer points''财产的helly-type定理。
A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in $\mathbb{R}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We prove fractional and colorful versions of a longstanding conjecture by Bárány, Katchalski, and Pach. We also show that a Minkowski norm admits an exact Helly-type theorem for diameter if and only if its unit ball is a polytope and prove a colorful version for those that do. Finally, we prove Helly-type theorems for the property of ``containing $k$ colinear integer points.
