The Marcus–de Oliveira conjecture predicts that for normal matrices A , B ∈ C n × n , the value det ( A + B ) lies in the convex hull of the n ! quantities ∏ i = 1 n ( λ i ( A ) + λ σ ( i ) ( B ) ) , σ ∈ S n . While the conjecture is classical and is known in the Hermitian case (Fiedler) and in several other structured settings, it remains open for general normal matrices. We develop a variational framework on the unitary group U ( n ) for the determinantal range W ↦ f ( W ) = det ( D A + W D B W ⁎ ) and analyze its support functionals. This approach unifies and reproves the conjecture for multiple positive classes, including Hermitian pairs, normal pairs with collinear spectra, and structured subclasses such as circulant/skew–circulant Toeplitz pairs and normal tridiagonal pairs admitting a common reducing block decomposition into essentially Hermitian blocks. In these settings, extremizers can be chosen monomial (permutation–phase) unitaries. We also identify an intrinsic obstruction to extending this first–order strategy to arbitrary normal matrices: support functional maximizers need not be monomial, as illustrated by an explicit 2 × 2 example.
Juan Ignacio Mulero-Martínez (Sun,) studied this question.