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
SN - 0012-365X
VL - 268
SP - 327
EP - 335
JO - Discrete Mathematics
JF - Discrete Mathematics
IS - 1-3
ER -