A Complex Analogue of Spencer's Six Standard Deviations Theorem and the Complex Banach-Mazur Distance

We investigate a complex analogue of Spencer's Six Standard Deviations Theorem. Specifically, we propose the following conjecture: for any dimension $n \geq 2$, given vectors $a_1, \ldots, a_n \in \mathbb{C}^n$ satisfying $\|a_i\|_{\infty} \leq 1$ for each $i=1, \ldots, n$, there exists a vector $x \in \mathbb{C}^n$ with all coordinates of modulus one such that $|\langle x, a_i \rangle| \leq \sqrt{n}$ for every $i=1, \ldots, n$. The bound of $\sqrt{n}$ is sharp, as demonstrated by the row vectors of any complex $n \times n$ Hadamard matrix. Furthermore, if the conjecture holds in dimension $n$, it implies that the Banach--Mazur distance between the complex $\ell_1^n$ and $\ell_{\infty}^n$ spaces is equal to $\sqrt{n}$. We prove the conjecture for $n =2, 3$, thereby establishing also that $d_{BM}(\ell_1^n, \ell_{\infty}^n) = \sqrt{n}$ for these dimensions. Additionally, we propose a conjecture about the Banach--Mazur distances between complex $\ell_p^n$ spaces and we verify it for $n=2$. This leads to a complete determination of all possible Banach--Mazur distances between complex $\ell_p^2$ spaces.