Saddle representations of positively homogeneous functions by linear functions

摘要

We say that a positively homogeneous function admits a saddle representationby linear functions iff it admits both an inf-sup-representation and asup-inf-representation with the same two-index family of linear functions. Inthe paper we show that each continuous positively homogeneous function can beassociated with a two-index family of linear functions which provides itssaddle representation. We also establish characteristic properties of thosetwo-index families of linear functions which provides saddle representations offunctions belonging to the subspace of Lipschitz continuous positivelyhomogeneous functions as well as the subspaces of difference sublinear andpiecewise linear functions.