Cross-positive matrices revisited

For a closed, pointed n-dimensional convex cone K in Rⁿ, let π(K) denote the set of all n × n real matrices A which as linear operators map K into itself. Let ∑(K) denote the set of all n × n matrices that are cross-positive on K, and L(K) = ∑(K) ∩ [- ∑(K)], the lineality space of ∑(K). Let Λ = RI, the set of all real multiples of the n × n identity matrix I. Then. π(K)+Δ⊆π(K)+L(K)⊆cl[π(K)+Δ]=Σ(K). The final equality was proved in 1970 by Schneider and Vidyasagar, who showed also that π(K) + Λ = ∑(K) when K is polyhedral but not when K is a three-dimensional circular cone. They asked for a general characterization of those K for which the equality holds. It is shown here that if n ≥ 3 and the cone K is strictly convex or smooth, then π(K) + Λ ≠ ∑(K); hence for n ≥ 3 the equality fails for "almost all" K in the sense of Baire category. However, the equality does hold for some nonpolyhedral K, as was shown by a construction that appeared in the third author's 1977 dissertation and is explained here in more detail. In 1994 it was shown by Stern and Wolkowicz that the weaker equality π(K) + L(K) = ∑(K) holds for all ellipsoidal (as well as for all polyhedral) K, and they wondered whether this equality holds for all K. However, their equality certainly fails for all strictly convex or smooth K such that L(K) = Λ, and it is shown here that this also includes "almost all" K when n ≥ 3.