TITLE: Minimizing the excursion of a sum of unit vectors
QUESTION [16 upvotes]: I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose
sum is zero:
$$ v_1 + v_2 + \cdots + v_n = 0 \; .$$
Now I form the closed polygon $P$ in space by placing them head to tail.
So the vertices of $P$ are 
$$ 0, v_1, (v_1+v_2), \ldots, (v_1+\cdots +v_{n-1}), 0 \; .$$
My question is:

What is the minimum excursion from the origin achievable by
  shuffling the vectors by a permutation of $(1,2,\ldots,n)$?

For example, the $12$ red vectors below wander $\sqrt{10}$
from the (purple) origin, but the light blue vectors—the same in  a different order—stay within $\sqrt{3}$.

          


(These particular vectors derive from the vertices of a
cuboctahedron, so some are 
negations of others.)
Is there some constant $r_{\min}$ independent of $n$ such that the sum
can always be arranged to be at most $r_{\min}$,
i.e., lie within an origin-centered ball of that radius?  Or must $r_{\min}$ depend on $n$?
Is there some natural algorithm for minimizing the excursion, or must I
(in the worst case)
try all $n!$ permutations?
Of course the same question can be asked in any dimension $\mathbb{R}^d$, but my focus is $\mathbb{R}^3$.
Thanks for ideas and/or pointers!
Update1.
The suggestion (in the comments) that $r_{\min} = \sqrt{d}$ in $\mathbb{R}^d$, based on an answer to
the previous MO question, "Bounding a signed sum of complex numbers,"
is intriguing, and may be true.  But I do not see that it is proved by that answer.
Update2.
With key phrases suggested by Nik Weaver, I found a 1981 paper by Imre Bárány, "A Vector-Sum Theorem and its Application to Improving Flow Shop Guarantees" (Math. Oper. Res. link), which shows that $r_{\min} < \frac{3}{2} d$.

REPLY [12 votes]: This is a famous open problem, $r_{\min}\le d$ is known as the Steinitz-lemma. It is conjectured that $r_{\min}= O(\sqrt d)$ but even $r_{\min}= o(d)$ is open. See also http://www.renyi.hu/~barany/cikkek/steinitz.pdf , section 3.