Abstract

Let EE be a Hausdorff locally convex space, E′E^\prime its dual, and X⊂EX\subset E a proper convex cone, not necessarily closed. Recall that an open ray δ\delta of XX is said to be extreme if (x∈δ(x \in \delta and x=y+zx = y +z with y,z∈X∖0)y,z \in X \setminus 0) implies (y,z∈δ)(y,z \in \delta). We denote by E(X)\mathcal{E}(X) the union of all extreme open rays of XX. We say that XX has the property of integral representation if for each x∈Xx \in X there is a positive Radon measure mm on E(X)\mathcal{E}(X) such that each f∈E′f \in E^\prime is mm-integrable and m(f)=f(x)m(f)=f(x).
Recently E. Thomas ["Integral representations in conuclear cones", J. Convex Analysis 1/2 (1994) 225–258] proved a theorem of integral representation for a class of convex cones, called conuclear. The aim of this work is to give a quite different presentation of his results, with the help of other tools, one of which is new (the pseudo-caps), allowing to avoid some of his hypotheses.

Author information

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R. Becker

U.R.A.754 au C.N.R.S., Université Pierre et Marie Curie-Paris 6, Tour 46-0 – Boite 186, 4 Place Jussieu, 75252 Paris cedex 05, France

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Suggested citation

R. Becker. “A New Tool in the Theory of Integral Representation.” Journal of Convex Analysis 3 (1996), No. 2, 349–360. https://doi.org/10.68381/jca03023

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Published by Heldermann Verlag, 1996. Rights now held by Banach Press.