Miranda and Thompsonʹs trace inequality and a log convexity result

A geometric proof of Miranda and Thompsonʹs trace inequality is given, via Thompsonʹs singular-value-diagonal-element inequalities. Miranda and Thompsonʹs trace inequality is associated with the unitary group. We then deal with the cases associated with the orthogonal group and the special unitary group. We also discuss the convexity of the set of the diagonal elements of complex matrices with fixed singular values and determinant. Some question are asked. A log convexity result related to Gram-Schmidt decomposition is obtained.