On the corners of certain determinantal ranges (Q996223)

Let \(A\) be a complex \(n\times n\) matrix and \(\text{SO}(n)\) be the group of \(n\times n\) real orthogonal matrices of determinant \(1\). The authors consider the set \(\Delta(A)\subseteq\mathbb{C}\) consisting of all \(\det(A\circ Q)\) (\(Q\in \text{SO}(n)\)) where \(\circ\) denotes the Hadamard product. For each permutation \(\sigma\in S_{n}\) they define \(z_{\sigma}:=\prod_{i=1}^{n} a_{i\sigma(i)}\), and note that \(z_{\sigma}=\det(A\circ Q)\) for suitable signed permutation matrices \(Q\in \text{SO}(n).\) A (weakened) version of the Oliveira Marcus conjecture [see \textit{G. N. De Oliveira}, Linear Multilinear Algebra 12, 125--138 (1982; Zbl 0485.15006)] states that \(\Delta(A)\) is contained in the convex hull of the \(z_{\sigma}\) (\(\sigma\in S_{n}\)). The present authors provide some evidence towards the truth of this conjecture by showing that the local shape of \(\Delta(A)\) at \(z_{\sigma}\) is a truncated cone whose angle at \(z_{\sigma}\) equals the angle \(\widehat{z_{\sigma\tau}z_{\sigma}z}_{\sigma\tau^{\prime}}\) for some transpositions \(\tau,\tau^{\prime}\).