TITLE: Least norm in convex set in Banach space
QUESTION [10 upvotes]: The following statement is true for Hilbert spaces $H$:

Every closed convex set ${\cal C} \subset H$ has a unique element $x$ such that for any $y \in C$, we have $|x| \leq |y|$.

Is this statement still true for Banach spaces? If not, what is a counterexample?

REPLY [7 votes]: This is a bit too long for a comment. The question was answered by Jonas and Giuseppe. Let me add that a result sometimes called the Day-James theorem characterizes the spaces for which the statement in your question is true:

For a normed space $X$ the following are equivalent:

$X$ is strictly convex and reflexive (in particular $X$ is complete).
Every non-empty closed convex set in $X$ has a unique point of minimal norm.


See the discussion after the proof of Corollary 5.1.19 on page 436 of Megginson's book An introduction to Banach space theory.
The space in Giuseppe's answer is not strictly convex and the ones in Jonas's answer are neither strictly convex nor reflexive.
The question Distance minimizers in $L^1$ and $L^{\infty}$ could also be of interest to you.