Every $proximinal$ set must be $closed$, but the opposite is not true. I'm looking for such an example. A $closed$ set that is not $proximinal$ set ?
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The empty set is a trivial example.
In $l^1$ consider the set $$ C=\text{ cl conv }\left( \frac{n+1}ne_n \right)_{n\in\mathbb N}. $$ Then $dist(C,0)\le1$ as $\|\frac{n+1}ne_n\|_{l^1}\to1$ for $n\to\infty$.
Take $x\in C$. Then there is a sequence $(\lambda_n)$ of non-negative numbers with $\sum_{n=1}^\infty \lambda_n=1$ and $\sum_{n=1}^\infty \lambda_n \frac{n+1}ne_n =x$. Then we compute $$ \|x\|_{l^1} =\sum_{n=1}^\infty \lambda_n \frac{n+1}n=1+\sum_{n=1}^\infty \lambda_n \frac{1}n. $$ This proves $\|x\|_{l^1}\ge 1$. Moreover, there is an index $k$ such that $\lambda_k>0$. Hence $\|x\|_{l^1}\ge 1+\frac1k>1$. So there is no $x\in C$ of minimal distance to the origin. And $C$ is not proximinal.
daw
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The canonical example runs as follows.
Example. Let $\varphi$ be a continuous linear functional for which the supremum $\sup_{\lVert x \rVert \leq 1} |\varphi(x)|$ is not attained. By scaling, we may assume that $\lVert \varphi \rVert = 1$. Set $C = \varphi^{-1}(1)$ and note that this is a closed convex set in $X$ (in fact, it is an affine hyperplane). We claim that $C$ does not contain a point of minimal norm (i.e. a point closest to $0$).
Let $\{x_n\}_{n=1}^\infty$ be a sequence of vectors such that $\lVert x_n \rVert \leq 1$ and $|\varphi(x_n)| \geq 1 - \frac{1}{n}$ for all $n \in \mathbb{N}_1$. After multiplying with appropriate scalars, we may assume that $\varphi(x_n)$ is real and positive for all $n \in \mathbb{N}_1$, so $\varphi(x_n) \in [\frac{n - 1}{n},1)$. This shows that $\alpha_n x_n \in C$ for a sequence $\{\alpha_n\}_{n=1}^\infty$ of real scalars with $\alpha_n \in (1,\frac{n}{n-1}]$ for all $n$. Therefore $d(0,C) \leq 1$.
Since the supremum $\sup_{\lVert x \rVert \leq 1} |\varphi(x)|$ is not attained, we have $|\varphi(x)| < \lVert \varphi \rVert = 1$ whenever $\lVert x \rVert \leq 1$, which shows that $C$ does not contain a vector of norm $\leq 1$. Therefore $d(0,C) = 1$, and $C$ does not contain a point closest to $0$.
A deep result of James [Jam64] shows that a Banach space is reflexive if and only if for every continuous linear functional $\varphi$ the supremum $\sup_{\lVert x \rVert \leq 1} |\varphi(x)|$ is attained. Thus, we can find a counterexample in every non-reflexive Banach space.
It is not hard to write down concrete examples. For instance, let $X = c_0$ (the closed subspace of $\ell^\infty$ consisting of sequences that converge to $0$), and let $\varphi \in X' = \ell^1$ be given by any sequence $(\varphi_n)_{n=1}^\infty \in \ell^1$ such that $\varphi_n \neq 0$ for infinitely many values of $n$. If $x = (x_n)_{n=1}^\infty \in X$ with $\lVert x \rVert_\infty \leq 1$, then $|x_n| \leq 1$ for all $n \in \mathbb{N}_1$, and $\lim_{n\to\infty} x_n = 0$. Hence there exists some $N_0$ such that $|x_n| < 1$ for all $n \geq N_0$. Since $\varphi_n \neq 0$ for some $n \geq N_0$, it follows that $$ |\varphi(x)| = \left| \sum_{n=1}^\infty x_n\varphi_n\right| \leq \sum_{n=1}^\infty |x_n \varphi_n| < \sum_{n=1}^\infty |\varphi_n| = \lVert \varphi \rVert. $$ Therefore $\varphi$ does not attain its maximum on the unit ball, so it follows from the preceding example that $\varphi^{-1}(1) \subseteq c_0$ is not proximinal. Similar examples can be found in other non-reflexive spaces. (Exercise: find a closed convex subset $C \subseteq \ell^1$ which is not proximinal.)
As a final remark, I should point out that it is relatively easy to show that every non-empty closed convex set in a reflexive Banach space is proximinal (use the argument from [Meg98, Corollary 5.1.19]), so a convex counterexample to your question can only exist in a non-reflexive space. Furthermore, Blatter [Bla76] gave an example which shows that every non-complete normed vector space contains a closed convex subset which is not proximinal. Putting these results together, we get:
Theorem (James–Blatter). A normed vector space $X$ is reflexive if and only if every non-empty closed convex subset of $X$ is proximinal.
References.
[Jam64]: R. C. James, Characterizations of reflexivity. Studia Mathematica 23(3):205–216, 1964. https://doi.org/10.4064/sm-23-3-205-216 (open archive)
[Bla76]: Jörg Blatter, Reflexivity and the existence of best approximations. In: G. G. Lorentz, C. K. Chui, and L. L. Schumaker (eds.), Approximation Theory II. Academic Press, New York, 1976. Pages 299–301. https://archive.org/details/DTIC_ADA033032 (archived copy)
[Meg98]: Robert E. Megginson, An Introduction to Banach Space Theory. Graduate Texts in Mathematics 183, Springer, New York, 1998.
