Generalized convexity cones

Zvi Ziegler

Vol. 17 (1966), No. 3, 561–580

DOI: 10.2140/pjm.1966.17.561

Abstract

We consider in this paper generalized convexity cones C(ψ₁,⋯,ψ_n) with respect to an Extended Complete Tchebycheffian system {ψ₁(x),⋯,ψ_n(x)}. These cones play a significant role in various areas of mathematics, such as moment theory, theory of approximation and interpolation, and theory of differential inequalities.

The properties of the cone C(ψ₁,⋯,ψ_n) are investigated. In particular, the extreme ray problem is solved explicitly for this cone, and for the intersection of such cones. Several structural properties of the cones are then determined.

The cone dual to C(ψ₁,⋯,ψ_n), which was introduced by S. Karlin and A. Novikoff is examined and a characterization of the cones which are dual to intersections of generalized convex cones is given. Some extensions of known theorems are given as applications.

Mathematical Subject Classification

Primary: 26.52

Milestones

Received: 17 January 1965

Revised: 20 March 1965

Published: 1 June 1966