Existence and uniqueness of projections on closed convex sets

Question

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.

Answer 1

Take $R^2$ with norm $\|(x,y)\| = \max \{|x| , |y| \}$ it is clearly reflexive space.

now consider the line $x = 1$ in $R^2$ as our convex set $C$, and the point $(0 , 0)$. Then all points in the form $(1 , y)$ with $-1 \leq y \leq 1$ on the line serve as nearest point to $(0 , 0)$.

So being reflexive in not enough. You have to seek conditions explicitly on the norm of the space, topological behavior is not enough. I think norm has to be strictly convex.

Answer 2

Answering my own question since I found a good reference.

Let $X$ be a normed space and $C\subseteq X$. Then $C$ is called

• A set of uniqueness iff for every $x\in X$ there is at most one $c\in C$ with $d(x,C)=d(x,c)$
• A set of existence iff for every $x\in X$ there is $c\in C$ with $d(x,C)=d(x,c)$
• A Chebyshev set iff it is both a set of uniqueness and a set of existence

We have the following results:

Theorem: Let $X$ be a normed space, then the following are equivalent

• $X$ is strictly convex
• Every nonempty convex set in $X$ is a set of uniqueness
• Every nonempty closed convex set in $X$ is a set of uniqueness

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Theorem: Let $X$ be a normed space, then the following are equivalent

• $X$ reflexive
• Every nonempty closed convex set in $X$ is a set of existence

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Theorem (Day-James): Let $X$ be a normed space, then the following are equivalent

• $X$ reflexive and strictly convex
• Every nonempty closed convex set in $X$ is a Chebyshev set

All of these results can be found in Megginson's "An Introduction to Banach Space Theory" as theorem 5.1.17, corollary 5.1.19 and subsequent remarks.