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

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