Vector balancing is the problem of assigning signs to a given collection of vectors so that their signed sum has a small prescribed vector norm. A permutation variant also allows the coordinates of each vector to be independently permuted (Niles-Weed et al. 2026).
Niles-Weed et al. (2026) proved that, for any vectors 
, 
, ..., 
with Euclidean norm
at most 1, there are signs 
and coordinate permutations 
such that
 |
where 
is the infinity norm. The choices can be made online,
using only the vectors seen so far. The bound tends to
1 as 
and is asymptotically optimal, since a single coordinate unit
vector forces a bound of at least 1. The result concerns the variant with permutations, not sign choices alone.
