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?