The maximum number of ways to stabn convex nonintersecting sets in the plane is 2n−2

Abstract

LetS be a collection ofn convex, closed, and pairwise nonintersecting sets in the Euclidean plane labeled from 1 ton. A pair of permutations % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca% GGOaGaamyAamaaBaaaleaacaaIXaaabeaakiaacYcacaWGPbWaaSba% aSqaaiaaikdaaeqaaOGaaiilaiaac6cacaGGUaGaaiOlaiaacYcaca% WGPbWaaSbaaSqaaiaad6gacqGHsislcaaIXaaabeaakiaacYcacaWG% PbWaaSbaaSqaaiaad6gaaeqaaOGaaiilaiaacMcacaGGSaGaaiikai% aadMgadaWgaaWcbaGaamOBaaqabaGccaGGSaGaamyAamaaBaaaleaa% caWGUbGaeyOeI0IaaGymaaqabaGccaGGSaGaaiOlaiaac6cacaGGUa% GaaiilaiaadMgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamyAamaa% BaaaleaacaaIXaaabeaakiaacYcacaGGPaaacaGL7bGaayzFaaaaaa!5937! $$\left\{ {(i_1 ,i_2 ,...,i_{n - 1} ,i_n ,),(i_n ,i_{n - 1} ,...,i_2 ,i_1 ,)} \right\}$$ is called ageometric permutation of S if there is a line that intersects all sets ofS in this order. We prove thatS can realize at most 2n−2 geometric permutations. This upper bound is tight.