I have come across the following challenging problem in my analysis course: Let $K$ be a compact convex set in a normed linear space. Suppose that $\sup_{x,y\in K}\{||x-y||\}=\delta>0.$
Show there exists $x_0\in K$ such that $\sup_{y\in K}\{||x_0-y||\}<\delta$
We were given the following hint: since $K$ is compact, choose $a,b\in K$ such that $||a-b||=\delta$. Let $K_0$ be a maximal subset of $K$ containing $a$ and $b$ such that $||x-y||$ is either $0$ or $\delta$ whenever $x,y\in K_0$.
Now, I've made this progress so far: The set $K_0$ as described above is a discrete subset of a compact set; thus, it must be finite. My strategy so far has thus been: suppose, towards a contradiction that I can't find such an $x_0$. Then, since $K$ is compact, this means that for any $x\in K$, there is some $y\in K$ such that $||x-y||=\delta$. Using this, and convexity, I am then trying to show that I can add another point to $K_0$. I've experimented with just assuming for a bit that $K_0$ only has two elements and playing around with what this give me using the triangle inequality, but I haven't had much luck.
If anyone has some ideas on how to tackle this, it'd be much appreciated.