An Improved Vector Balancing Constant via the Hellinger Affinity

Let v1, . . . , vn be vectors in R
m of Euclidean norm at most one. The Koml´os signing prob-
lem asks for signs ϵj ∈ {−1, 1} such that the signed sum has small ℓ∞ norm, with a bound
independent of m and n. A recent result of Guo, Fang, and Lu establishes such a bound with
constant 3√
2π ≈ 7.5199, via a directional total variation invariant and a stability lemma whose
threshold is κ∥v∥2 ≤ 1/3. The threshold is a consequence of a height identity that controls the
mass retained by a symmetrized lift of the density under the Banaszczyk transform, together
with a linear estimate on the translation distance of the density. We replace the linear estimate
with a sharper bound derived from the exact Hellinger affinity of the product cosine-squared
density, combined with the linear bound. This gives an improved constant C
′ ≈ 7.4858. The
residual gap to the asymptotic value C
⋆ ≈ 6.9013 of the method is characterized by a Gaus-
sian comparison conjecture for localized product experiments; we prove the case k = 1 of that
conjecture in the range used by the reduction.