TAILIEUCHUNG - Báo cáo toán học: " Large convexly independent subsets of Minkowski sums"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Large convexly independent subsets of Minkowski sums. | Large convexly independent subsets of Minkowski sums Konrad J. Swanepoel Department of Mathematics London School of Economics and Political Science WC2A 2AE London UK Pavel Valtr Department of Applied Mathematics and Institute for Theoretical Computer Science Charles University Malostranske nam. 25. 118 00 Praha 1 Czech Republic valtr@ Submitted Aug 15 2009 Accepted Oct 21 2010 Published Oct 29 2010 Mathematics Subject Classifications Primary 52C10 Secondary 52A10. Abstract Let Ed n be the maximum number of pairs that can be selected from a set of n points in Rd such that the midpoints of these pairs are convexly independent. We show that E2 n Q. nỵ logn which answers a question of Eisenbrand Pach Rothvofi and Sopher 2008 on large convexly independent subsets in Minkowski sums of finite planar sets as well as a question of Halman Onn and Rothblum 2007 . We also show that L1 n2_ E3 n 3n2 O n3 2 . Let Wd n be the maximum number of pairwise nonparallel unit distance pairs in a set of n points in some d-dimensional strictly convex normed space. We show that W2 n 0 E2 n and for d 3 that Wd n 1 1 -7M n2 where a d G N 2V 2 d 2 a d is related to strictly antipodal families. In fact we show that the same asymptotics hold without the requirement that the unit distance pairs form pairwise nonparallel segments and also if diameter pairs are considered instead of unit distance pairs. 1 Three related quantities A geometric graph is a graph with the set of vertices in Rd and with each edge represented as a straight line segment between its incident vertices. Halman et al. 8 studied geometric Swanepoel gratefully acknowledges the hospitality of the Department of Applied Mathematics Charles University Prague. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R146 1 graphs for which the set of midpoints of the edges are convexly independent . they form the vertex set of their convex hull. For any finite set P c Rd let E P be the maximum .

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