tailieunhanh - Báo cáo toán học: "Convexly independent subsets of the Minkowski sum of planar point sets"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Convexly independent subsets of the Minkowski sum of planar point sets. | Convexly independent subsets of the Minkowski sum of planar point sets Friedrich Eisenbrand1 Janos Pach2 Thomas Rothvoh1 and Nir B. Sopher3 1 Institute of Mathematics Ecole Polytechnique Federate de Lausanne 1015 Lausanne Switzerland @ 2Courant Institute NYU and City College CUNY USA pach@ 3 School of Electrical Engineering Tel-Aviv University Tel-Aviv 69978 Israel sopherni@ Submitted Dec 14 2007 Accepted Mar 17 2008 Published Mar 20 2008 Mathematics Subject Classification 52C10 52A10 Abstract Let P and Q be finite sets of points in the plane. In this note we consider the largest cardinality of a subset of the Minkowski sum S c P Q which consist of convexly independent points. We show that if P I m and IQ I n then S I O m2 3 n2 3 m n . 1 Introduction In connection with a class of convex combinatorial optimization problems Onn and Roth-blum 2004 Halman et al. 2007 raised the following question. Given a set X of n points in the plane what is the maximum number of pairs that can be selected from X so that the midpoints of their connecting segments are convexly independent that is they form the vertex set of a convex polygon In the special case when the elements of X themselves are convexly independent they found a linear upper bound 5n 6 on this quantity. They asked whether there exists a subquadratic upper bound in the general case. In this note we answer this question in the affirmative by establishing an upper bound of O n4 3 . We first reformulate the question in a slightly more general form. Let P and Q be sets of size m and n in the plane. The Minkowski sum of P and Q is P Q p q I p 2 P q 2 Qg. What is the maximum size of a convexly independent subset of P Q THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N8 1 More precisely we would like to estimate the function M m n which is the largest cardinality of a convexly independent set S which is a subset of the Minkowski sum of some planar point .