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Let $K$ be a convex set in $\mathbb{R^n}$

a) For arbitrary $x_1,x_2,...,x_{n+1}\in K$ prove that intersection of all sets $\frac{1}{n}x_i+K$ is nonempty for all $i=1,2,...,n+1$.

b) If $K$ is compact set, prove that there exists $x\in \mathbb{R^n}$ such that $x-\frac{1}{n}K \subseteq K$

In a) part I tryed to use induction to prove the claim but not sure if it is correct way. Is there any other posibility to prove? In part b) I am not sure how the assumption that $K$ is compact can be used? Should I conclude something about b) part based on the a) part?

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  • $\begingroup$ a) Start studying the sequence of points in $K$, if you look $\frac{x_i}{n}\in K$, $\forall i$. $\endgroup$
    – james watt
    Nov 18, 2018 at 12:41

1 Answer 1

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(a): The intersection contains $\frac1n(x_1+\dots+x_{n+1}).$

(b): Use (a) and Helly's theorem to deduce that there exists some $x$ in $\bigcap_{k\in K}(\frac1nk+K).$ Unwrapping notation shows that $x-\frac1n k\in K$ for each $k\in K,$ which is what you want.

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  • $\begingroup$ Thank you so much for help. $\endgroup$
    – XYZ
    Nov 25, 2018 at 11:54

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