The regular simplex maximization conjecture for minimum projective distance

Let $\mathbf{u}_0,\dots,\mathbf{u}_n$ be unit vectors representing the vertices of a simplex in complex projective space, in general position, and let

$$D=|\det(\mathbf{u}_0,\dots,\mathbf{u}_n)|.$$

Let $d_{\min}$ denote the minimum of the projective distances from the vertices to their opposite hyperplanes. Fix $0<D\leq 1$ and consider all configurations with this value of $D$. Regular simplex conjecture. Among all such configurations, the configuration with the largest $d_{\min}$ will be a regular simplex. The preceding theorem gives the bound $d_{\min}^n\leq D$, while equality of the relevant distances alone is not sufficient for equality; the conjecture identifies the regular simplex as the optimizer under fixed determinant magnitude.