Bang vectors in the real polarization problem: The exact dimensional range

Source abstract

We determine the exact dimensional range in which a normalized longest signed sum, which we call a Bang vector, always satisfies the real polarization inequality: this prescription is universally valid if and only if $n\leq 14$. For every $n\geq 15$ we construct $n$ unit vectors whose all-positive sum is, up to global sign, the unique longest signed sum, while its normalized direction has polarization product smaller than $n^{-n/2}$. The counterexamples are given by a single explicit family, and for the same configurations we exhibit nonmaximal signed sums whose normalized directions do satisfy the polarization bound.