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On Geometric Permutations Induced by Line Transversals through a Fixed Point #
 

Summary: On Geometric Permutations Induced by Line Transversals
through a Fixed Point #
Boris Aronov + Shakhar Smorodinsky #
June 5, 2004
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 long­standing 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) =
# d
i=0 # n-1
i # = #(n d ) is the number of pairs
of antipodal cells in a simple arrangement of n great (d - 1)­spheres in a d­sphere. By a similar
argument, we show that the maximum number of connected components of the space of all line
transversals through a fixed point to a family of n > 2 possibly intersecting convex objects is

  

Source: Aronov, Boris - Department of Computer and Information Science, Polytechnic University

 

Collections: Computer Technologies and Information Sciences