Some projective distance inequalities for simplices in complex projective space (Q2352937)

The authors prove the following main result: Let \(u_0,\ldots, u_n\) be \(n+1\) linearly independent unit vectors in \(\mathbb C^{n+1}\) representing \(n+1\) linear forms defining \(n+1\) hyperplanes in general position in \(\mathbb{CP}^n\), which we think of as the faces of a projective simplex. For each \(j\) from \(0\) to \(n\), let \(d_j\) denote the Fubini-Study distance from the hyperplane presented by \(u_j\) to the opposite vertex of the simplex. Let \(d_{\min}\) denote the minimum of the \(d_j\) (and \(d_{\min}^n\) its \(n\)-th power). Then \[ d_{\min}^n\leq|\det(u_0,\dots, u_n)|\leq\,d_{\min}. \]