On geometric permutations induced by lines transversal through a fixed point Conference Paper uri icon

abstract

  • A line transversal of a family S of n pairwise disjoint convex objects is a straight line meeting all members of S. A geometric permutation of S is the pair of orders in which members of S are met by a line transversal, one order being the reverse of the other. In this note we consider a long-standing open problem in transversal theory, namely that of determining the largest number of geometric permutations that a family of n pairwise disjoint convex objects in R d can admit. We settle a restricted variant of this problem. Specifically, we show that the maximum number of those geometric permutations to a family of n> 2 pairwise disjoint convex objects that are induced by lines passing through any fixed point is between K (n-1, d-1) and K (n, d-1), where K (n, d)= Σ di= 0 (n-1/i)= Θ (nd) is the number of pairs of antipodal cells in a simple arrangement of n great (d-1)-spheres in a d-sphere. By a similar …

publication date

  • January 1, 2005