Upper bounds on geometric permutations for convex sets

Abstract

LetA be a family ofn pairwise disjoint compact convex sets inR d. Let % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdy0aaS% baaSqaaiaadsgaaeqaaOGaaiikaiaad2gacaGGPaGaeyypa0JaaGOm% aiabfo6atnaaDaaaleaacaWGPbGaeyypa0JaaGimaaqaaiaadsgacq% GHsislcaaIXaaaaOWaaeWaaeaadaqhaaWcbaGaiaiG0caaamyAaaqa% aiaad2gacqGHsislcaaIXaaaaaGccaGLOaGaayzkaaaaaa!4A12! $$\Phi _d (m) = 2\Sigma _{i = 0}^{d - 1} \left( {_i^{m - 1} } \right)$$ . We show that the directed lines inR d, d ≥ 3, can be partitioned into % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0de9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdy0aaS% baaSqaaiaadsgaaeqaaOWaaeWaaeaadaqadaqaamaaDaaaleaacaaI% YaaabaGaamOBaaaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa!3CFF! $$\Phi _d \left( {\left( {_2^n } \right)} \right)$$ sets such that any two directed lines in the same set which intersect anyA′⊆A generate the same ordering onA′. The directed lines inR 2 can be partitioned into 12n such sets. This bounds the number of geometric permutations onA by 1/2φ d ford≥3 and by 6n ford=2.