On properties of univariate max functions at local maximizers

Abstract More than three decades ago, Boyd and Balakrishnan established a regularity result for the two-norm of transfer function at maximizers. Their extends easily to statement that maximum eigenvalue univariate real analytic Hermitian matrix family is twice continuously differentiable, with Lipschitz second derivative, all local maximizers, property useful in several applications we describe. We also investigate whether this smoothness max functions more generally. show pointwise finite set q -times differentiable must have zero derivative maximizer $$q=1$$ q = 1 , but arbitrarily close maximizer, may not be defined, even when $$q=3$$ 3 isolated.