Large Signed Sums and the Polarization Constant of Convex Bodies

Abstract

As a counterpart to classical vector balancing, we study large signed sums of unit vectors in finite-dimensional Minkowski spaces and develop their connection with polarization problems for convex bodies. For every convex body $K\subset\mathbb{R}^d$ containing the origin in its interior and every number $n\geq1$ of vectors, we show that the corresponding large signed sum and polarization constants coincide in a common quantity $\mu(K,n)$, which depends only on the symmetric core $K\cap(-K)$. Our main result determines the sharp universal lower bound \[ \mu(K,n) \geq \mu(B_\infty^d,n) = \frac{1}{n}\left\lceil\frac{n}{d}\right\rceil, \] identifying parallelotopes as global minimizers for every $d$ and $n$. In the opposite direction, we prove general upper bounds that show, in particular, that the Euclidean ball is a maximizer up to an absolute constant factor. We also obtain sharp and nearly sharp results in several special cases, including the planar setting. As $n\to\infty$, we prove that $\mu(K,n)$ converges to the Macphail constant of $K$, equivalently to its $1$-absolutely summing constant. Thus, the finite signed-sum problem provides a discrete counterpart of classical Banach space invariants. Combining this connection with sharp results on projection constants, we characterize equality in the corresponding upper bound for the Macphail constant in terms of maximal real equiangular tight frames. Finally, our methods extend to arbitrary vector families, support functions of compact convex sets, and circumradii and diameters of Minkowski sums.