1. ## Convex set proof

Hello,

I'm trying to write a proof which is, in my point of view, easy to figure geometrically but not so easy to write it. I hope you can help me.

Let $\displaystyle E$ be a vector space and $\displaystyle u,v \in E$. The line segment that starts from $\displaystyle u$ and ends in $\displaystyle v$ is the set $\displaystyle [u,v] = \{(1-t)u+tv; 0\leq t \leq 1\}$. A set $\displaystyle X\subset E$ is called convex whenever $\displaystyle u,v\in X \Rightarrow [u,v]\subset X$.

Prove that the intersection $\displaystyle X_1\cap\ldots\cap X_m$ of convex sets $\displaystyle X_1,\ldots,X_m \subset X$ is a convex set.

First, I supposed that if we have $\displaystyle X_a \subset X_b \subset \ldots \subset X_m$, the intersection will be $\displaystyle X_a\cap\ldots\cap X_m = X_a$ for any a, b in the naturals. As $\displaystyle X_a$ is convex, the intersection will be too.

Second, I supposed that if $\displaystyle X_1\cap\ldots\cap X_m = \{v\}$ then by definition $\displaystyle [v,v] = \{v\} \subset X_1\cap\ldots\cap X_m$. Is this correct?

Then I don't know exactly what I do from know: Should I suppose the intersection a set with {u,v}, and so on ({u,v,w}, ...)? I don't know if I'm going through the right direction.
I appreciate if you can help me or give me an advice about how to get confident in this kind of problem.

2. Originally Posted by bondesan
Hello,

I'm trying to write a proof which is, in my point of view, easy to figure geometrically but not so easy to write it. I hope you can help me.

Let $\displaystyle E$ be a vector space and $\displaystyle u,v \in E$. The line segment that starts from $\displaystyle u$ and ends in $\displaystyle v$ is the set $\displaystyle [u,v] = \{(1-t)u+tv; 0\leq t \leq 1\}$. A set $\displaystyle X\subset E$ is called convex whenever $\displaystyle u,v\in X \Rightarrow [u,v]\subset X$.

Prove that the intersection $\displaystyle X_1\cap\ldots\cap X_m$ of convex sets $\displaystyle X_1,\ldots,X_m \subset X$ is a convex set.

First, I supposed that if we have $\displaystyle X_a \subset X_b \subset \ldots \subset X_m$, the intersection will be $\displaystyle X_a\cap\ldots\cap X_m = X_a$ for any a, b in the naturals. As $\displaystyle X_a$ is convex, the intersection will be too.

Second, I supposed that if $\displaystyle X_1\cap\ldots\cap X_m = \{v\}$ then by definition $\displaystyle [v,v] = \{v\} \subset X_1\cap\ldots\cap X_m$. Is this correct?

Then I don't know exactly what I do from know: Should I suppose the intersection a set with {u,v}, and so on ({u,v,w}, ...)? I don't know if I'm going through the right direction.
I appreciate if you can help me or give me an advice about how to get confident in this kind of problem.

I think it's way easier: let $\displaystyle u,v \in X_1\cap\ldots\cap X_n\Longrightarrow [u,v]\in X_1\cap\ldots\cap X_n$ since $\displaystyle [u,v]\in X_i\,,\,\,\forall\,\,i=1,2,...,n$ .

Tonio