Question

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 $$\sum_{j=1}^n\alpha_{j}\beta_{\sigma(j)},\qquad\sigma\in {\mathfrak S}_n.$$

Answer 1

This claim is nothing but the well-known Marcus and de Oliveria conjecture, which has been open since 1973 or earlier.

Reference: Open Problems in Matrix Theory, X. Zhan.

For the simpler case of Hermitian matrices, the claim holds; a slightly more general case seems to be the paper "The validity of the Marcus-de Oliveira conjecture for essentially Hermitian matrices."

PS: you might want to add the "open problem" tag to your question.