Generalizations of topheavy cones

Abstract

A closed convex cone Kin a finite-dimensional real inner product space Vis said to be symmetric relative to subspace Lof V, or L-symmetric, if x− y∈ Kwhenever x∈ L y∈ L ⊥and x+ y∈ K. Fiedler and Haynsworth have shown that a full pointed cone Kis symmetric relative to a one-dimensional subspace Liff it is “top heavy” relative to some norm ν on L ⊥, i.e., it has the form {x 1 e 1+y| x 1⩾ v(y)}for some e∈ K∩ L. This result is first extended to arbitrary L-symmetric cones using extended seminorms. Cones top heavy relative to vectorial norms are discussed. Finally, it is shown that the cone of positive operators on a given L-symmetric cone is itself symmetric relative to a subspace of operators determined by L.