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Thread: Convex set proof

  1. #1
    Junior Member bondesan's Avatar
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    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.
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  2. #2
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    Quote Originally Posted by bondesan View Post
    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
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