Non-uniqueness of closest point in a closed, convex set

Question

I have seen that if $X$ is a Hilbert space and $K$ is some non-empty, closed, convex subset then for every $x \in X $ there exists a unique $y \in K$ which is closer to any other point of $K$. However, I wanted to see when uniqueness does not hold when making the weaker assumption that X is reflexive. Any suggestions?

Answer 1

Take $\mathbb R^2$, with the norm $\bigl\lVert(x,y)\bigr\rVert_1=\lvert x\rvert+\lvert y\rvert$. Then the line segment $K$ going from $(1,0)$ to $(0,1)$ is non-empty, closed and convex. However, each element of $K$ is at distance $1$ from $(0,0)$.