Convexity of Point Set

Sandip Das (sandipdas@isical.ac.in)Indian Statistical Institute

Convex SetA 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, . . ., tn with i

n ti =1

such that a = in ti.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.)

income

Queries: Find person having maximum income

Find person whose expenditure is minimum

Find person having maximum savings

expen

dit

ure 1

23

5

7

8

Linear ProgrammingMaximizing

c1 x1+ c2 x2+ . . .+ cn xn

Subject to a11 x1+ a12 x2+ . . .+ a1n xn ≤ b1

a21 x1+ a22 x2+ . . .+ a2n xn ≤ b2

…

an1 x1+ an2 x2+ . . .+ ann xn ≤ bn

Linear Programming (Contd.)

a11 x1+ a12 x2+ . . .+ a1n xn = b1

is a hyperplane in n dimensional plane

a11 x1+ a12 x2+ . . .+ a1n xn ≤ b1

Implies a halfplane bounded by this

hyperplane

feas

ible

sol

utio

n

Linear Programming (Contd.)Set of constraints generate intersection of n

hyperplanes

Intersection of convex regions is convex

Hyperplane co

rresp

onding to

optimiza

tion cr

iteria

by setti

ng

functional v

alue as some

consta

nt.

Linear Programming (Contd.)Set of constraints generate intersection of n

hyperplanes

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 lLeft

side of l

right side of l

Center point (Contd.)Consider a point x in 2DDraw a line l through x.We can compute # of points on left with

respect to lSimilarly # 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

xx

x

Can you identify a point x such thatc(x) ≤1

Centerpoint TheoremA 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, …, Cn be 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 TheoremConsider 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 centerpointShreesh Maharaj 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 centerpoint region 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 setA 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 TheoremG(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 TheoremGiven 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-HoleLet 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….