The polyhedral Lagrange principle

Abstract

This note is a supplement to [1]. We give a criterion for inhomogeneous consequences of simultaneous polyhedral sublinear inequalities. Using it, we prove a version of the Lagrange principle for dominated polyhedral sublinear operators. Also, we fill a gap in the proof of Theorem 3.4 in [1] about simultaneous inhomogeneous complex operator inequalities. This article uses the terminology and notation of [1]. Lemma. Let X be a real vector space. Assume that p1, . . . , pN ∈ PSub(X) := PSub(X,R) and p ∈ Sub(X). Assume further that v ∈ R and u1, . . . , uN ∈ R make consistent the simultaneous sublinear inequalities pk(x) ≤ uk, with k := 1, . . . , N . The following are equivalent: (1) {p ≥ v} ⊃ N ⋂