On the irreducibility, Lyapunov rank, and automorphisms of special Bishop–Phelps cones

Motivated by optimization considerations, we consider cones in RnRn – to be called special Bishop–Phelps cones – of the form {(t,x):t≥||x||}{(t,x):t≥||x||}, where ||⋅||||⋅|| is a norm on Rn−1Rn−1. We show that when n≥3n≥3, such cones are always irreducible. Defining 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 l1l1-cone (which is a special Bishop–Phelps cone with 1-norm), we show that any lplp-cone, for 1≤p≤∞1≤p≤∞, p≠2p≠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 l1l1-cone.