# The determinant of the sum of normal matrices

Given two normal matrices $A,B\in M_n({\mathbb C})$ whose respective spectra are $(\alpha_{1},\ldots,\alpha_{n})$ and $(\beta_{1},\ldots,\beta_{n})$, is it true that $\det(A+B)$ belongs to the convex hull of the set of numbers $$\prod_{i=1}^n(\alpha_i+\beta_{\sigma(i)}),$$ as $\sigma$ runs over the set ${\mathfrak S}_n$ of permutations of $\{1,\ldots,n\}$ ?

Nota. It is known (see Exercise 101) that the trace of $AB$ belongs to the convex hull of the set of numbers $$\sum_{j=1}^n\alpha_{j}\beta_{\sigma(j)},\qquad\sigma\in {\mathfrak S}_n.$$

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Interesting question, I hope some one will provide a solution. –  Betrand Jun 29 '12 at 11:45