Note on the degree of concavity of the determinant on sets of positive definite quadratic forms (Q1094443)

For \((a_{ij})\) a real symmetric positive \(n\times n\)-matrix, let \(D(a_{ij})\) denote its determinant. A result of Minkowski says that \(D^{1/n}\) is concave on the space \({\mathcal P}_ n\) of all \(n\times n\)- matrices (considered as a convex cone in \({\mathbb{R}}^{n(n+1)/2}).\) Let \({\mathcal P}^*_ n\) denote the subspace of \({\mathcal P}_ n\) consisting of all \((a_{ij})\in P_ n\) with \(a_{11}=...=a_{nn}=1\) and let \({\mathcal R}^*_ n\) denote the subset of all Minkowski reduced matrices in \({\mathcal P}^*_ n\). For a convex subset E of \({\mathcal P}_ n\) let \(\nu\) (E) be the (unique) real number such that \(D^{\alpha}\) is concave on E iff \(\alpha\leq \nu (E).\) It is shown that \(\nu\) (\({\mathcal P}_ n)=1/n\), \(\nu\) (\({\mathcal P}^*_ n)=1/(n-1)\), \(\nu\) (\({\mathcal R}^*_ 2)=5/2\), \(\nu (R^*_ 3)=1\), \(1/(n-2)\leq \nu ({\mathcal R}^*_ n)\leq (n+3)/n(n-1)\) for \(n\geq 4\). This continues earlier work of \textit{C. E. Nelson} [Aequationes Math. 11, 163- 168 (1974; Zbl 0291.10022)] and is related to inequalities of Ky Fan and Bergström for positive definite matrices. Originally the question arose from a result of \textit{B. L. van der Waerden} [ibid. 2, 233-247 (1969; Zbl 0174.083)] on reduction theory.