On a minimax equality for seminorms Original Research Article

For seminorms short parallel·short parallel, short parallel·short parallel0, and short parallel·short parallel1, defined on a real or complex vector space X and induced by positive semidefinite Hermitian forms, we present two different proofs of the equality imagesupvarkappaset membership, variantX1varkappamax≤1varkappa = min0≤t≤1supvarkappaset membership, variantX1varkappat≤1 varkappa, where varkappamax = max{varkappa0, varkappa1} and varkappa2t = (1 − t)varkappa20 + tvarkappa21, t set membership, variant [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.