Let $X$ be a strictly convex Banach space, let $C\subseteq X$ be a nonempty closed convex set, and let $P\_C$ be the set-valued *metric projection*
$$P\_C(x) = \{y\in C : \|x-y\| = d(x,C)\} . $$
We know that since $X$ is strictly convex, the metric projection onto a closed convex set is either empty or a singleton.


My question: Does anyone know of examples where $P\_C(x)$ is empty? (assuming $X$ is a strictly convex Banach space and C is a nonempty closed convex set)



Example of empty projection in strictly convex Banach space
To complete Narutaka OZAWA's answer in comment by a concrete example as asked in the OP, here is a bounded linear functional on $c\_0$ not attaining its norm w.r.to (an equivalent) strictly convex norm.


On the space $c\_0 $ consider a norm $\Vert x\Vert :=\Vert x\Vert \_\infty +\Vert x\Vert \_2$, which is obtained adding a (weighted) pre-Hilbert norm, hence strictly convex  

$$\Vert x\Vert \_2:=\sqrt{\sum\_{k=1}^\infty 2^{-k}x\_k^2}$$
to the standard norm $$\Vert x\Vert\_\infty:=\max\_{k\ge1}|x\_k|.$$ Thus $ \Vert x\Vert \_\infty\le \Vert x\Vert\le 2 \Vert x\Vert \_\infty$, so $\|\cdot\|$ is equivalent to $\|\cdot\|\_\infty$, and it is strictly convex.


Consider the linear form $f\in(c\_0)^\*=\ell\_1$ defined by $f\_k:=2^{-k}$. Then for any $x\in c\_0\setminus\{0\}$
$$\langle f,x\rangle=\sum\_{k=1}^\infty 2^{-k}x\_k\le\sqrt{\sum\_{k=1}^\infty 2^{-k}}\sqrt{\sum\_{k=1}^\infty 2^{-k}x\_k^2}=\|x\|\_2 $$
and also
$$\langle f,x\rangle<\Big( \sum\_{k=1}^\infty 2^{-k}\Big)\|x\|\_\infty=\|x\|\_\infty =\|x\|-\|x\|\_2.$$
Therefore $\displaystyle\langle f,x\rangle<\frac12\|x\|$ for all $x\neq 0$. On the other hand, for the sequence $x^n:=\sum\_{k=1}^ne\_k=({1,\dots,1},0,0\dots)\in c\_0$ one has
$$\frac{\langle f,x^n\rangle}{\|x^n\|}=\frac{ \sum\_{k=1}^n 2^{-k}}{1+\sqrt{\sum\_{k=1}^n 2^{-k}}}=\frac{ 1- 2^{-n}}{1+\sqrt{1- 2^{-n}}}=\frac12+o(1).$$
So $f$ has dual norm $\displaystyle \frac12$, but it is not attained on the closed unit ball.


And the affine closed hyperplane $C:=\{f=\frac12\}$ has no element of minimum norm (minumum distance from $0$)


