Unlike the minimum eigenvalue, which is concave, the minimum singular value is the difference of two convex functions, and is neither concave nor convex for matrix dimension >= 2. See section 5 of “On Extreme Singular Values of Matrix Valued Functions”, L. Qi and R.S. Womersley, Journal of Convex Analysis, volume 3 (1996), available at http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.299&rep=rep1&type=pdf . A simple 2 by 2 counter-example to the concavity of the minimum singular value is presented in Remark 5.2.