The sharp exponent for the Banach--Mazur distance to the cube

Let R_∞(n) be the maximal Banach--Mazur distance from an n-dimensional real normed space to _∞ⁿ. We prove that R_∞(n)≥ cn2/3(log(2n))⁻¹⁰. Together with the upper bound of Friedland, Naor, and Youssef, this gives R_∞(n) = n2/3+o(1). The examples are polars of symmetric Gaussian polytopes with $2n$ generators, and the lower bound holds with exponentially high probability. A joint minimization over coefficient representations and positive weights yields an exact second-variation identity and sparse coefficients. A Gaussian projection argument controls all low-rank completions of the recorded matrix. An adaptive choice of exceptional generators then reduces the remaining case to a fixed-shift Gaussian small-ball estimate over two finite nets.