Axiomatization of the index of pointedness for closed convex cones

Abstract

Abstract. Let C(H ) denote the class of closed convex cones in a Hilbert space H . One possible way of measuring the degree of pointedness of a cone K is by evaluating the distance from K to the set of all nonpointed cones. This approach has been explored in detail in a previous work of ours. We now go beyond this particular choice and set up an axiomatic background for addressing this issue. We define an index of pointedness over H as being a function f : C(H ) → R satisfying a certain number of axioms. The number f (K ) is intended, of course, to measure the degree of pointedness of the cone K . Although several important examples are discussed to illustrate the theory in action, the emphasis of this work lies in the general properties that can be derived directly from the axiomatic model. 47L07, 52A20. Mathematical subject classification:

Deterministic and stochastic methods for computing volumetric moduli of convex cones

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Norm-Induced Densities and Testing the Boundedness of a Convex Set

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