TY - JOUR

T1 - On neighbors in geometric permutations

AU - Sharir, Micha

AU - Smorodinsky, Shakhar

N1 - Funding Information:
Work on this paper has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the US–Israeli Binational Science Foundation, by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski—MINERVA Center for Geometry at Tel Aviv University.

PY - 2003/7/6

Y1 - 2003/7/6

N2 - We introduce a new notion of 'neighbors' in geometric permutations. We conjecture that the maximum number of neighbors in a set S of n pairwise disjoint convex bodies in R(d) is O(n), and we prove this conjecture for d=2. We show that if the set of pairs of neighbors in a set S is of size N, then S admits at most O(N d-1) geometric permutations. Hence, we obtain an alternative proof of a linear upper bound on the number of geometric permutations for any finite family of pairwise disjoint convex bodies in the plane.

AB - We introduce a new notion of 'neighbors' in geometric permutations. We conjecture that the maximum number of neighbors in a set S of n pairwise disjoint convex bodies in R(d) is O(n), and we prove this conjecture for d=2. We show that if the set of pairs of neighbors in a set S is of size N, then S admits at most O(N d-1) geometric permutations. Hence, we obtain an alternative proof of a linear upper bound on the number of geometric permutations for any finite family of pairwise disjoint convex bodies in the plane.

KW - Geometric permutations

KW - Line transversals

UR - http://www.scopus.com/inward/record.url?scp=84867930712&partnerID=8YFLogxK

U2 - 10.1016/S0012-365X(03)00052-9

DO - 10.1016/S0012-365X(03)00052-9

M3 - Article

AN - SCOPUS:84867930712

VL - 268

SP - 327

EP - 335

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -