A SANDWICH THEOREM, THE MOMENT PROBLEM, FINITE-SIMPLICIAL SETS, AND SOME INEQUALITIES

摘要

We recall the following sandwich type problem. Let X be a convex subset of a vector space E, f, g : X → R two maps, g convex, f concave, g ≤ f. The problem is: under what conditions upon X, for any pair (g, f) as above, there exists h : X → R affine, such that g ≤ h ≤ f. When X is a compact Hausdorff topological space and S is a convex cone of lower bounded, lower semicontinuous functions on X for which axiom S_1 (p. 493) is fulfilled, the problem is solved in [5], p. 505. We show that for any finite-simplicial subset X, the sandwich problem stated above has at least one solution (we say that a convex subset X of a vector space E is finite-simplicial if and only if for any finite dimensional compact convex subset K is contained in X, there exists a finite dimensional simplex T such that K is contained in T is contained in X). The novelty here is the fact that X is not supposed to be bounded in a locally convex topology. The idea of the proof is to reduce the problem to the particular case when X is a finite dimensional simplex. As applications of our results, we deduce some inequalities for positive numbers and for self-adjoint bounded operators acting on a Hilbert space, pointing out the relationship between scalar and operator inequalities (see Corollary 3.9). In Section 4 we solve special moment problems in spaces of differential functions on [0, b] and a space of functions on the unit sphere of R~n.