# Proving Finite Convex Sets Intersection is Convex

• retspool
In summary, the task is to prove that the intersection of a number of finite convex sets is also a convex set. A set is considered convex if for any points x and y in the set and any value a between 0 and 1, the point ax + (1-a)y is also in the set. The solution involves correcting the definition of convex set and then using this property to show that adding multiple points in the intersection satisfies the same condition, proving that the intersection is also a convex set.
retspool

## Homework Statement

Prove that the intersection of a number of finite convex sets is also a convex set

## Homework Equations

I have a set is convex if there exists x, y in the convex S then

f(ax + (1-a)y< af(x) + (1-a)y

where 0<a<1

## The Attempt at a Solution

i can prove that
f(ax + (1-a)y) < f(x) given that x is a global minimizer

then i guess that i could find another arbritary point close to x , x_1, x_2 and add their given function satisfying the convex condition to get

Sum f(axi + (1-a)y) < Sumf(xi) where i= 1, 2,...nany help would be appreciated

Your definition of "convex set" is wrong; there is no function involved. A set $$S$$ (in some real vector space $$V$$) is convex if, whenever $$x, y \in S$$ and $$0 \leq a \leq 1$$, then also $$ax + (1 - a)y \in S$$.

Once you correct that, if you find yourself working too hard, you're doing something wrong. Just chase the definitions.

## 1. What is a finite convex set?

A finite convex set is a set of points that lie within a convex shape, where any two points within the set can be connected by a straight line that lies entirely within the shape.

## 2. How do you prove that the intersection of two finite convex sets is also convex?

To prove that the intersection of two finite convex sets is convex, we need to show that for any two points within the intersection, the line connecting them lies entirely within the intersection. This can be done by showing that the intersection of the two sets contains all the points on the line segment connecting the two points.

## 3. Can you provide an example to illustrate the proof of finite convex sets intersection being convex?

Yes, consider two finite convex sets, A and B, where A is a triangle and B is a square. The intersection of A and B will be a smaller triangle that is contained within both A and B. Any two points within this smaller triangle can be connected by a straight line that lies entirely within the triangle, proving that the intersection is convex.

## 4. Are there any special cases where the intersection of finite convex sets may not be convex?

Yes, there are special cases where the intersection of finite convex sets may not be convex. One example is when the two sets have a common point but do not overlap in any other way. In this case, the intersection would just be a single point, which is not considered a convex shape.

## 5. Why is it important to prove that the intersection of finite convex sets is convex?

Proving that the intersection of finite convex sets is convex is important because it allows us to make assumptions and draw conclusions about the properties of the intersection. This can be useful in various fields such as optimization, geometry, and computer science.

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