The Geometry of Real Anisotropic Bohnenblust--Hille Constants

We determine the growth scale of the optimal constants in the real anisotropic Bohnenblust–Hille inequality. For an exponent vector ${\mathbf{q}}^{(m)}$, write $C_{{\mathbf{q}}^{(m)}}^{(m)}$ for its optimal constant and $d_m$ for its diameter. These constants are superpolynomial precisely when $m{d}_m\mathrm{/}\log m\to \mathrm{\infty }$; throughout this regime, their logarithm has the sharp scale $m{d}_m$. If also $d_m\to 0$, then $\log C_{{\mathbf{q}}^{(m)}}^{(m)}\mathrm{/}(m{d}_m)$ lies asymptotically in the interval $[\frac{\log 2}{4},\frac{2-\log 2-\gamma }{4}]$, where $\gamma$ is the Euler–Mascheroni constant; this interval has width less than ${10}^{-2}$. We also solve the extremal problems at fixed diameter and fixed total deficit. The normalized reciprocal-deficit profiles order the canonically arranged optimal constants by majorization, yield exact formulas on a full-dimensional region, and recover the lower coefficient $(\log 2)\mathrm{/}4$ throughout a broad class of anisotropic regimes.