Combinatorial Geometry 邓俊辉 清华大学计算机系 年10月12日星期一 2015年10月12日星期一 2015年10月12日星期一 2015年10月12日星期一 2015年10月12日星期一 2015年10月12日星期一下午12时18分 下午12时18分 下午12时18分 下午12时18分 下午12时18分

2 Junhui Deng, Tsinghua Computer Radon's Theorem ☞ Radon's partition Radon partition –Given P a family of sets in E d, if there are two disjoint, non-empty subfamilies P 1 and P 2 of P such that conv(P 1 )  conv(P 2 )  , then (P 1, P 2 ) is called a Radon partition of P ☞ Radon's Theorem –Every family of n  d+2 sets in E d admits a Radon partition

3 Junhui Deng, Tsinghua Computer Radon's Theorem ☞ Kirchberger's Theorem –For any Radon partition (P 1, P 2 ) of a family P of n  d+2 sets in E d, there is a subfamily U  P with (dim(P 1  P 2 )+2) sets such that (P 1  U, P 2  U) is a Radon partition of U (and hence of P) ☞ Tverberg's Theorem –  Every set of (m-1)(d+1)+1 points in E d can be divided into m (pairwise disjoint) subsets whose convex hulls have a common point; –  the number (m-1)(d+1)+1 is the smallest which has the stated property

4 Junhui Deng, Tsinghua Computer Helly's Theorem ☞ [Helly, 1923] –[Finite version] A family of finite convex sets admits a nonempty common intersection iff each of its (d+1)-cardinality subfamilies does –[Infinite version] A family of infinite compact convex sets admits a nonempty common intersection iff each of its (d+1)-cardinality subfamilies does

5 Junhui Deng, Tsinghua Computer Transversal ☞ k-Transversal –Given F a family of sets in E d, a k-flat T is called a k-transversal of F if T meets every member of F ☞ Examples –0-transversal / Helly theorem –1-transversal / stabbing line

6 Junhui Deng, Tsinghua Computer The space of k-transversals ☞ Given F a family of convex sets in E d –the topological space of all k-flats intersecting F is called the space of k- transversals of F ☞ The space of 0-transversals –convex (why?) ☞ The space of k-transversals (k>=1) –not convex –even not connected

7 Junhui Deng, Tsinghua Computer Finding k-Transversal ☞ Goal –not just a single k-transversal –a data structure representing the entire space of k-transversals ☞ Problems –How much time/space is needed for the construction? –What's the complexity of such a data structure? –What's the combinatorial complexity of this space?

8 Junhui Deng, Tsinghua Computer Ham-Sandwich Theorem ☞ [Discrete version] –Let P 1, …, P d be d finite sets of points in E d –There exists a hyperplane h that simultaneously bisects P 1, …, P d

9 Junhui Deng, Tsinghua Computer Minkowski's first Theorem ☞ [Minkowski, 1891] –Let C  E d be symmetric, convex, bounded, and suppose that vol(C) > 2 d –Then C contains at least one lattice point different from 0 ☞ Ex: Regular Forest –Diameter of forest = 26m –Diameter of trees = ? ?

10 Junhui Deng, Tsinghua Computer Two-Square Theorem & Four-Square Theorem ☞ [Two-square Theorem] –Each prime p  1 (mod 4) can be written as a sum of two squares –i.e.,  prime p = 4k + 1,  a, b  Z s.t. p = a 2 + b 2 –e.g. 13 = = ☞ [Four-square Theorem] –Any natural number can be written as a sum of 4 squares of integer

11 Junhui Deng, Tsinghua Computer Four-Square Theorem ☞ 2004 –= (25, 25, 23, 15) = (27, 25, 19, 17) = (27, 25, 23, 11) = (27, 25, 25, 5) = (28, 26, 20, 12) –= (28, 28, 20, 6) = (29, 21, 19, 19) = (29, 25, 23, 3) = (30, 28, 16, 8) = (31, 23, 17, 15) –= (31, 27, 17, 5) = (31, 29, 11, 9) = (31, 31, 9, 1) = (32, 20, 18, 16) = (32, 24, 20, 2) –= (32, 28, 14, 0) = (32, 30, 8, 4) = (33, 23, 19, 5) = (33, 25, 13, 11) = (33, 25, 17, 1) –= (33, 29, 7, 5) = (34, 24, 16, 4) = (34, 28, 8, 0) = (35, 21, 13, 13) = (35, 21, 17, 7) –= (35, 23, 13, 9) = (35, 23, 15, 5) = (35, 27, 5, 5) = (35, 27, 7, 1) = (36, 16, 16, 14) –= (36, 26, 4, 4) = (37, 17, 15, 11) = (37, 19, 15, 7) = (37, 21, 13, 5) = (37, 23, 9, 5) –= (37, 25, 3, 1) = (38, 20, 12, 4) = (39, 17, 13, 5) = (39, 19, 11, 1) = (40, 14, 12, 8) –= (40, 16, 12, 2) = (40, 18, 8, 4) = (40, 20, 2, 0) = (41, 11, 11, 9) = (41, 15, 7, 7) –= (41, 17, 5, 3) = (43, 9, 7, 5) = (43, 11, 5, 3) = (44, 6, 4, 4) = (44, 8, 2, 0)

12 Junhui Deng, Tsinghua Computer The Erdos-Szekeres Theorem ☞ Convex Independent Set –A set X  R d is called convex independent if for every x  X, we have x  conv(X\{x}) ☞ [Erdos & Szekeres, 1935] –For every k  N, there exists a number n(k) such that any n(k)-point set X  R 2 in general position contains a k-point convex independent subset ☞ What is the lower bound of n(k)? –Ex: n(5)  9

13 Junhui Deng, Tsinghua Computer The k-Set Problem ☞ k-Sets –Let P be a configuration of n points in E d and let S be a halfspace k-set –P  S is called a k-set of P if card(P  S) = k, 0  k  n ☞ Problems –What are the upper/lower bounds for the maximum number of k-sets for all configurations of n points in E d, in terms of n and k? –What are the upper/lower bounds for the maximum number of k-sets for all configurations of n points in E 2, in terms of n and k?

14 Junhui Deng, Tsinghua Computer The k-Set Problem ☞ Lower bounds –  (nlog(k+1))Edelsbrunner & Welzl, 1985 –  (ne sqrt(log(k+1)) )Toth, 1999 ☞ Upper bounds – O(n*sqrt(k+1))Lovasz, 1971 – O(n*sqrt(k+1)/log * (k+1))Pach et al., 1989 – O(n*cbrt(k+1))Dey, 1998