combinatorial geometry 邓俊辉清华大学计算机系 [email protected] deng...

Combinatorial Geometry

邓俊辉

清华大学计算机系[email protected]

http://vis.cs.tsinghua.edu.cn:10020/~deng

2023年4月19日下午 10:16

mailto:[email protected]?subject=Tsinghua%2000240074@2003Spring

http://vis.cs.tsinghua.edu.cn:10020/~deng

22Junhui Deng, Tsinghua Computer

Radon's Theorem

☞ Radon's partition

– Given P a family of sets in Ed, if there are two disjoint, non-empty subfamilies P1 and P2 of P such that conv(P1) conv(P2) , then (P1, P2) is called a Radon Radon

partitionpartition of P

☞ Radon's Theorem

– Every family of n d+2 sets in Ed admits a Radon partition


Radon's Theorem

☞ Kirchberger's Theorem

– For any Radon partition (P1, P2) of a family P of n d+2 sets in Ed, there is a subfamily U P with (dim(P1P2)+2) sets such that (P1U, P2U) is a Radon partition of U (and hence of P)

☞ Tverberg's Theorem

Every set of (m-1)(d+1)+1 points in Ed 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


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


Transversal

☞ k-Transversal

– Given F a family of sets in Ed, 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


The space of k-transversals

☞ Given F a family of convex sets in Ed

– 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


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?


Ham-Sandwich Theorem

☞ [Discrete version]

– Let P1, …, Pd be d finite sets of points in Ed

– There exists a hyperplane h that simultaneously bisects P1, …, Pd


Minkowski's first Theorem

☞ [Minkowski, 1891]

– Let C Ed be symmetric, convex, bounded, and suppose that vol(C) > 2d

– Then C contains at least one lattice point different from 0

☞ Ex: Regular Forest

– Diameter of forest = 26m

– Diameter of trees = ??


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 = a2 + b2

– e.g. 13 = 22 + 32

17 = 12 + 42

☞ [Four-square Theorem]

– Any natural number can be written as a sum of 4 squares of integer


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)


The Erdos-Szekeres Theorem

☞ Convex Independent Set

– A set X Rd 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 R2 in general position contains a k-point convex independent subset

☞ What is the lower bound of n(k)?

– Ex: n(5) 9


The k-Set Problem

☞ k-Sets

– Let P be a configuration of n points in Ed and let S be a halfspace

– PS is called a k-setk-set of P if card(PS) = 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 Ed, 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 E2, in terms of n and k?


The k-Set Problem

☞ Lower bounds

(nlog(k+1)) Edelsbrunner & Welzl, 1985

(nesqrt(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

combinatorial geometry 邓俊辉 清华大学计算机系 [email protected] deng...

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combinatorial geometry 邓俊辉清华大学计算机系 [email protected] deng...