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- W2020981363 abstract "For a closed, pointed n -dimensional convex cone K in R n , 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 = R I , 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." @default.
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- W2020981363 date "1995-07-01" @default.
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- W2020981363 title "Cross-positive matrices revisited" @default.
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- W2020981363 doi "https://doi.org/10.1016/0024-3795(93)00364-6" @default.
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