This paper studies a new twist on a long-standing puzzle in convex geometry. Instead of only choosing plus or minus signs for a list of vectors, the authors allow, for each vector, a choice of sign and a permutation of its coordinates. They prove explicit bounds that control every partial sum (the “prefix” sums) at once, and they show those bounds are asymptotically optimal as the ambient dimension grows. The proof is purely geometric and constructive.

Concretely, the authors show that for any list of bounded vectors one can pick, for each vector, a sign and a permutation so that all partial sums remain small in norm. The choices can be made online: each sign and permutation is chosen one-by-one based only on what came before, using a greedy rule. The paper also gives exact examples showing the upper bound is attained in small dimensions (for instance, in dimension 2 by taking v1 = (1/2, −1/2) and v2 = (1/2, 1/2)).

The proof splits each input vector into two pieces: a part parallel to the all-ones vector and a part orthogonal to it. The orthogonal pieces can be balanced by permutations alone. That step uses the centered regular permutahedron — a convex polytope that is the convex hull of all coordinate permutations of a particular vector — together with the notion of majorization (which characterizes when one vector is a convex combination of permutations of another). The permutahedron can be realized as a projection of a cube (a zonotope), and this geometric picture lets the authors keep the partial sums inside a fixed polytope throughout the greedy construction.

The paper also extends the result beyond the usual maximum norm. In a more general lp-to-lq setting the authors prove a bound that grows like n^{1/q} in the ambient dimension n, and they show this growth rate cannot be improved in general. In some parameter ranges the authors obtain the exact leading constant and in other ranges they prove the dependence on n is sharp.