Dominated extensions of functionals and V-convex functions of cancellative cones (original) (raw)

CHARACTERIZING THE CONTINGENT CONE'S CONVEX KERNEL

Pure and Applied Functional Analysis, 2020

In this paper we present a direct characterization of the convex kernel of the contingent cone and use this result to sharpen the assumptions under which the calculus of the contingent and adjacent cones is known to be valid. We apply this calculus to develop necessary optimality conditions for a nonsmooth mathematical program.

Construction of a cone by using weak*- total families

Applied Mathematical Sciences, 2013

This paper is devoted to construct a cone by using the so called weak*-total families in different spaces (Banach space, Hilbert space, metric space,...etc.). We were advised some forms for weak*-total subsets to construct an exact forms for cones like (Normal cones, Allows plastering cones, Solid cones,...etc.).

Two characterizations of ellipsoidal cones

2012

We give two characterizations of cones over ellipsoids. Let C be a closed pointed convex linear cone in a finite-dimensional real vector space. We show that C is a cone over an ellipsoid if and only if the affine span of ∂C ∩ ∂(a − C) has dimension dim(C) − 1 for every point a in the relative interior of C. We also show that C is a cone over an ellipsoid if and only if every bounded section of C by an affine hyperplane is centrally symmetric.

Archimedean levels, semispaces, and majorization of convex cones

Archiv der Mathematik, 1993

Introduction. As the term is used here, a cone is a nonempty subset C of a real vector space such that 0r and C=C +C=]O, CO[C. In other words, C is a convex cone that has the origin 0 as an apex but does not contain the origin. Along with C, we consider its linear hull L c = C -C. We say that a cone C is majorized by a cone K, and we write C < m K, if there exists a linear transformation T: Lc ~ LK such that T(C) c K. Each such Tis called a majorization of C by K, and C and K are m-equivalent (written C ~ K) if each majorizes the other. The transformation T is not assumed to be continuous, and in fact (except in some supplementary remarks) the only topology involved here is the usual topology on finite-dimensional subspaces. This paper is concerned with the problems of recognizing when one given cone is majorized by another, and of classifying m-equivalence classes of cones. It provides a complete solution of these problems for finite-dimensional cones. Each cone C generates two partial orderings of its linear hull and these play an essential role. They are defined as follows for x, y ~ L c :

Lifts of Non-compact Convex Sets and Cone Factorizations

In this paper we generalize the factorization theorem of Gouveia, Parrilo and Thomas to a broader class of convex sets. Given a general convex set, we define a slack operator associated to the set and its polar according to whether the convex set is full dimensional, whether it is a translated cone and whether it contains lines. We strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone, but also its recession cone is the image of the linear slice of the closed convex cone. We show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.

Haar negligibility of positive cones in Banach spaces

St Petersburg Mathematical Journal, 2016

We discuss the Haar negligibility of the positive cone associated with a basic sequence in a separable Banach space. In particular, we show that up to equivalence, the canonical basis of c0 is the only normalized subsymmetric unconditional basic sequence whose positive cone is not Haar null, and the only normalized unconditional basic sequence whose positive cone contains a translate of every compact set. We also show that an unconditional basic sequence with a non–Haar null positive cone has to be c0-saturated in a very strong sense, and that every quotient of the space generated by such a sequence is c0-saturated.

Linear extension sums as valuations on cones

Journal of Algebraic Combinatorics, 2012

The geometric and algebraic theory of valuations on cones is applied to understand identities involving summing certain rational functions over the set of linear extensions of a poset. This paper presents a different viewpoint on the following two classes of rational function summations, which are both summations over the set L(P ) of all linear A. Boussicault · V. Féray ( ) LaBRI, Université Bordeaux 1,