Sharp constant for permutation balancing in the ell_infinity norm

Determine the sharp constant for permutation vector balancing when both the input and output norms are ell_infinity, namely for vectors in the ell_infinity unit ball balanced using signs and coordinate permutations while controlling all prefix sums in the ell_infinity norm.

Background

The paper studies vector balancing in which each vector may be assigned both a sign and a permutation of its coordinates, with the goal of controlling the maximum norm of all prefix sums. Theorem \ref{thm:komlos-permutations-pq} establishes an asymptotic bound of 3+o_n(1) when p=\infty and q=\infty.

The authors note that this constant may not be optimal. An explicit two-vector example shows that the optimal constant is at least 2, leaving the exact sharp constant for the \ell_\infty-to-\ell_\infty case unresolved.