Motivated by optimization considerations, we consider special Bishop-Phelps cones in R^n which are of the form {(t,x): t \geq ||x||} for some norm on R^(n-1). We show that for n bigger than 2, such cones are always irreducible. De fining the Lyapunov rank of a proper cone K as the dimension of the Lie algebra of the automorphism group of K, we show that the Lyapunov rank of any special Bishop-Phelps polyhedral cone is one. Extending an earlier known result for the l_1 cone (which is a special Bishop-Phelps cone with 1-norm), we show that any l_p cone, for p different from 2, has Lyapunov rank one. We also study automorphisms of special Bishop-Phelps cones, in particular giving a complete description of the automorphisms of the l_1 cone.
Citation
Research Report TRGOW13-01, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA. August 2013