Is it possible to partition a real Banach space into closed half-lines? I've been thinking about the Chebyshev conjecture in Hilbert spaces, and one consequence of the existence of a non-convex Chebyshev set is that one can partition the space into an (obviously uncountable) partition of closed half-lines ("partition" in the strong, set-based sense, not the disjoint interior topological sense).
(To see this, one can use the inversion method of Ficken and Asplund to form a uniquely remotal set, and the non-empty pre-images of the furthest point function form such a partition of the space.)
G. G. Johnson proved (with errors that were later corrected) in 1987 that there's a non-convex Chebyshev set in $c_{00}$, the space of finitely supported sequences, with the usual inner product. The inversion argument still holds without completeness, so $c_{00}$ has a uniquely remotal set with the inner product, and as such, it definitely does admit a partition into closed half-lines.
So, to answer the question in the title, all I'd need is a renorming of $c_{00}$ to make it a complete normed linear space. I guess the first question is, can such a renorming be done? Additionally, what if I insisted additionally that the result is a Hilbert space?
I'm also interested in the question in finite-dimensions. It is known that there are no non-convex Chebyshev sets in finite dimensional Hilbert spaces, so the same construction will not work. Clearly it doesn't work on the real line, but as soon as we get to two or more dimensions, it becomes less clear. Even on the plane, where the question seems it should be significantly simpler than in higher dimensions, there's still a frustrating lack of tools to work with. At best, I can partition the plane missing a single point (and I don't see "many" ways of doing that), but I can't see how to fill the whole thing, or even fill it while missing exactly two points!
If these questions spark any inspiration, I'd love it if you'd share it with me!
 A: So, years after positing this question, I've come up with a disappointing (and somewhat embarrassingly easy) answer to this question: it is possible to partition $\Bbb{R}^2$ with closed half-lines. Consequently, any real vector space, complete or incomplete, with dimension at least $2$, possibly infinite, can be partitioned with closed half-lines, as every such space can be partitioned into affine spaces of dimension $2$.
To construct such a partition, begin by partitioning the third quadrant of $\Bbb{R}^2$, i.e.
$$\{(x, y) \in \Bbb{R}^2 : x \le 0, y \le 0\}$$
with closed half-lines. There's no need to be fancy; we can simply partition it with half-lines of the form $\{(\alpha, 0) + t(0, -1) : t \ge 0\}$, where $\alpha$ ranges over $(-\infty, 0]$.
We then encroach on the rest of the plane by tiling with "pieces", each congruent to $(0, 1] \times (-\infty, 0]$. We can partition these pieces into closed half-lines $\{(\alpha, 0) + t(0, -1) : t \ge 0\}$, where $\alpha \in (0, 1]$. Note we can cover any congruent piece, since the set of half-lines is stable under rotations (and scaling, so even similarity is enough). In particular, we can also partition $(-\infty, 0] \times (0, 1]$.
Now, for $n \ge 0$, let
$$Q_n = \{(x, y) \in \Bbb{R}^2 : x \le n, y \le n\}.$$
We recursively construct a nested sequence of sets of halflines $P_n$ such that $P_n$ partitions $Q_n$. The set $P = \bigcup_n P_n$ will therefore form a partition of $\Bbb{R}^2$ into closed half-lines.
Let $P_0$ be any partition of $Q_0$, such as the one above. Assume that for some $n \ge 0$, we have a nested sequence of sets of closed half-lines $P_0 \subseteq \ldots \subseteq P_n$ defined that partition $Q_0, \ldots, Q_n$ respectively. Note that
$$Q_{n+1} = Q_n \cup (n, n+1] \times (-\infty, n] \cup (-\infty, n + 1] \times (n, n+1],$$
and the union is disjoint. We can therefore form $P_{n+1}$ by
\begin{align*}
P_{n+1} = P_n &\cup \{\{(\alpha, n) + t(0, -1) : t \ge 0\} : \alpha \in (n, n + 1]\} \\
&\cup \{\{(n + 1, \alpha) + t(-1, 0) : t \ge 0\} : \alpha \in (n, n + 1]\},
\end{align*}
which is a partition of $Q_{n+1}$ into half-lines. By induction, the claim holds true, and $P$ is a partition of $\Bbb{R}^2$ into closed half-lines.

There's a more tricky version of this question, where the endpoints of the closed half-lines form a bounded set. Johnson's example in $c_{00}$ also satisfies this condition, as will the particular hypothetical partition that I'm interested in. I don't have a counterexample satisfying this condition, but this is a question for another time. If you have any ideas, please comment and let me know!
