Let $X$ be a normed vector space. I'm interested in determining what are the minimal assumptions on $X$ that guarantee the existence and uniqueness of projections on closed convex sets and in counterexamples showing that those assumptions are indeed necessary.
In particular let P1 and P2 be the following statements
P1: (existence) for every convex closed set $C\subseteq X$ and every $x\in X$ there exist $y\in C$ with $\|x-y\|=d(x,C)$.
P2: (uniqueness) for every convex closed set $C\subseteq X$ and every $x\in X$ there exist a unique $y\in C$ with $\|x-y\|=d(x,C)$.
What I know so far is that P2 holds in all uniformly convex Banach spaces, while to get P1 is enough to assume that $X$ is a reflexive Banach space, but I don't have an example of a reflexive Banach space not satisfying P2 and I don't know if those assumptions can be further weakened.