Convexity of Point Set

Convexity of Point Set SandipDas (sandipdas@isical.ac.in) Indian Statistical Institute

Convex Set • A set C d is convex if for every two points a, b C, the line segment joining a and b is also contained in C.

Convex Set (Contd.) • A set C d is convex if for every two points a, b C, the line segment joining a and b is also contained in C. • A set C d is convex if for every two points a, b C, and for every t[0, 1], the point t.a + (1- t).b belongs to C. Are These Two Definitions Equivalent?

Convex hull • The convex hull CH (S) of a set S is the smallest convex set that contains S.

Convex hull (Contd.) • Intersection of all convex set containing S. Algebraic Observation: A point a belongs to CH (S)iff there exist points s1, s2, . . ., snS, and non-negative real number t1, t2, . . ., tnwith inti =1 such that a = inti.si =1 =1

Convex hull: Application in optimization Consider the following Database Person income expenditure … … … Queries:Find person having maximum income Find person whose expenditure is minimum Find person having maximum savings

Application in optimization (Contd.) expenditure income Queries:Find person having maximum income Find person whose expenditure is minimum Find person having maximum savings 5 2 3 8 7 1

Linear Programming Maximizing c1 x1+ c2 x2+ . . .+ cnxn Subject to a11 x1+ a12 x2+ . . .+ a1nxn≤ b1 a21 x1+ a22 x2+ . . .+ a2nxn≤ b2 … an1 x1+ an2 x2+ . . .+ annxn≤ bn

Linear Programming (Contd.) a11 x1+ a12 x2+ . . .+ a1nxn= b1 is a hyperplane in n dimensional plane a11 x1+ a12 x2+ . . .+ a1nxn≤ b1 Implies a halfplane bounded by this hyperplane feasible solution

Linear Programming (Contd.) Set of constraints generate intersection of nhyperplanes Intersection of convex regions is convex Hyperplane corresponding to optimization criteria by setting functional value as some constant.

Linear Programming (Contd.) Set of constraints generate intersection of nhyperplanes Intersection of convex regions is convex Intersection region may be empty => no solution Intersection region may be unbounded => it may generate unbounded optimal solution

Center point Looking for center point among points arranged on a line. Have a sense of center point but not clear - Mean ? -Median ?

Center point (Contd.) Observe that Median say x is such a point where | # of points on left of x- # of points on right of x| ≤1 We want to extend this idea in 2D n points in a plane.

Center point (Contd.) n points in a plane. Left and right is not well defined on plane. We can define left and right with respect to a line l Left side of l right side of l

Center point (Contd.) Consider a point x in 2D Draw a line lthrough x. We can compute # of points on left with respect to l Similarly # of points on right with respect to l So, | # of points on left of l - # of points on right of l | varies as the line l rotate and passing through x x l

Center point (Contd.) | # of points on left of l - # of points on right of l | What is the maximum value of this difference for all line l passing through x Let us say that value as c(x) x l

Center point (Contd.) The term c(x) may be considered as a measure of x for being a center Can you identify a point x such that c(x) is less than equal to 1? x

Center point (Contd.) For any point set Can you identify a point x such that c(x) is less than equal to 1? Does such a point always exist?

Center point (Contd.) Example: Let the point set be Can you identify a point x such that c(x) ≤1 x x x

Centerpoint Theorem A point xRd is called a centerpoint of a point set if each closed halfspace containing x contains at least n/(d+1) points of the point set. Theorem: Each finite point set in Rd has at least one centerpoint. Follows from Helly’s theorem.

Helly’s Theorem Let C1, C2, …, Cnbe convex sets in 2D plane. Suppose that the intersection of every 3 of these sets is nonempty. Then the intersection of all the Ci is nonempty.

Proof of Centerpoint Theorem Consider any point set with n points. Take all convex set containing at least 2n/3 points. Number of such convex sets are finite. Observe that intersection of any three of them is not null Hence, from Helly’s theorem, intersection of all such convex hull is not null. Any point on that intersection is the centerpoint.

Algorithm for finding centerpoint ShreeshMaharaj et al. proposed an excellent algorithm in O(n) time Prune and search technique T(n) = T(c.n) +O(n), 0< c <1 Generate a convex region such that centerpointregion of point set including vertices of convex region is a superset of earlier one If some vertices of that convex region is discarded centerpoint remains same. Discard that fraction of boundary points, and continue the process.

Convex independent set A set S Rd is convex independent if all points in S lie on convex hull of S That is for every xS, x conv{S\{x}} Let P be a set of points and the points be in general position. Any three point subset is convex independent But any subset of 4 points is not convex independent

Convex independent set (contd.) Suppose the set P contains 5 point May we always get a subset of size 4 that are convex independent? Size of convex hull will be either 3, 4 or 5 If the size of convex hull is 5, then … …

Ramsey Theorem G(V, E) is a graph with |V|=6, then either G or Gc must have a triangle. So, R(3, 3) = 6 If the number of vertices is sufficiently large, there always exist a k vertex subset Y such the all hyperedge of 4 vertices is in G or in Gc.

Erdös-Szekeres Theorem Given n points set, color a 4 tuple red if its 4 points are convex independent and blue otherwise. From Ramsey Theorem, there is a k point subset such that all hyperedge is same color. But for k≥ 5, this color cannot be blue. So, that k point subset is convex independent

Erdös-Szekeres Theorem (Contd.) For every natural number k, there exist a number n(k) such that any n(k) point set in the plane in general position contains a k-point convex independent subset. 2k-2 + 1 ≤ n(k) ≤ 2k-5Ck-2 - 2

K-Hole Let X be a set of point. A k-point set Y is called a k-hole in the point set if Y is convex independent and conv(Y) X = Y. Erdös raised the question about the size of point set for k-hole 3-hole? 4-hole? 5-hole? 6-hole? 7-hole … … Does not exist.

A lot of questions remain unanswered….

Convexity of Point Set