ON A MINIMAX EQUALITY FOR SEMINORMS

For seminorms \\.\\, \\.\\(0), and \\.\\(1), defined on a real or complex vector space X and induced by positive semidefinite Hermitian forms, we present two different proofs of the equality [GRAPHICS] where \\x\\(max) = max {\\x\\(0), \\x\\(1)} and \\x\\(2)(t) - (1 - t) \\x\\(2)(0) + t \\x\\(2)(1), t is an element of [0, 1]. During the course of the first proof, results on the geometry of the joint numerical range of two real-valued quadratic forms are given for spaces equipped with a semidefinite Hermitian form, which may be of independent interest. In the second proof, using a more direct approach, the minimax equality is first proved for finite-dimensional X and norms generated by inner products, and this result is then extended to the general case.