On the dual of a result of Miranda and Thompson

For A and B real matrices with prescribed singular values, Miranda and Thompson characterized max tr(AUBV) when U and V range over SO(n), the real proper orthogonal group. Motivated by this result, we investigate the location of det(A + UBV) for A, B, U, V as previously. The corresponding problem for A and B complex matrices and U and V ranging over the unitary group is also discussed. Analogies are drawn between the determinantal problems and their tracial versions.