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 $n\ge 2$, given vectors $a_1,\dots ,a_n\in {\mathbb{C}}^n$ satisfying $\mathrm{\parallel }{a}_i{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1$ for each $i=1,\dots ,n$, there exists a vector $x\in {\mathbb{C}}^n$ with all coordinates of modulus one such that $\mathrm{\mid }⟨x,a_i⟩\mathrm{\mid }\le \sqrt{n}$ for every $i=1,\dots ,n$. 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 ${\mathrm{\ell }}_1^n$ and ${\mathrm{\ell }}_{\mathrm{\infty }}^n$ spaces is equal to $\sqrt{n}$. We prove the conjecture for $n=2,3$, thereby establishing also that $d_{BM}({\mathrm{\ell }}_1^n,{\mathrm{\ell }}_{\mathrm{\infty }}^n)=\sqrt{n}$ for these dimensions. Additionally, we propose a conjecture about the Banach–Mazur distances between complex ${\mathrm{\ell }}_p^n$ spaces and we verify it for $n=2$. This leads to a complete determination of all possible Banach–Mazur distances between complex ${\mathrm{\ell }}_p^2$ spaces.