Math Proof Application to geometry What does it mean for a f

Math Proof

Application to geometry

What does it mean for a figure to be \"convex\"?

A figure is convex if any two points it contains are connected by a line

all of whose points fall within the figure.

HW 1) A half-plane is convex (all the points on one side of a line including the line.)

2) Any intersection of half-planes is convex.

3) Given any figure, I can consider the intersection of all the half-planes

that contain it. This is the smallest closed convex set contain the given

figure.   

\"Closed\" means any point lying outside the figure sits in a disk of positive

radius that is itself entirely outside the figure.

Solution

Given a half plane in R2R2 described by the equation 2x3y6, how would one go about proving this vector space \"SS\" is convex?

Let

H={x,yR2:2x3y6}H={x,yR2:2x3y6}

(1t)x+ty=(1t)x1+ty1,(1t)x2+ty2(1t)x+ty=(1t)x1+ty1,(1t)x2+ty2

2{(1t)x1+ty1}3{(1t)x2+ty2}=(1t)(2x13x2)+t(2y13y2)(1t)6+t6=62{(1t)x1+ty1}3{(1t)x2+ty2}=(1t)(2x13x2)+t(2y13y2)(1t)6+t6=6

Hence HH is convex.

2)In fact, the intersection of any number of convex shapes is again convex. This follows directly from the definition of convexity.

Thus a half plane is convex so intersection convex.