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?
(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.