Let $X$ be a  Banach  space, $Csubset X$  be  a  closed  convex  set  included  in  a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a  bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set  $C$,  so that ${mathop{rm int}}(mathrm{dom} sigma_C) neqem...

Meiotic chromosomes in an oocyte are not only a maternal genome carrier but also provide a positional signal to induce cortical polarization and define asymmetric meiotic division of the oocyte, resulting in polar body extrusion and haploidization of the maternal genome. The meiotic chromosomes play dual function in determination of meiosis: 1) organizing a bipolar spindle formation and 2) indu...

S. Sherman has shown [4] that if the self adjoint elements of a C* algebra form a lattice under their natural ordering the algebra is necessarily commutative. In this note we extend this result to real Banach algebras with an identity and arbitrary Banach * algebras with an identity. The central fact for a real Banach algebra A is that if the positive cone is defined to be the uniform closure o...

The set of Euclidean distance matrices has a well-known representation as a convex cone. The problems of representing the group averages of K distance matrices are discussed, but not fully resolved, in the context of SMACOF, Generalized Orthogonal Procrustes Analysis and Individual Differences Scaling. The polar (or dual) cone representation, corresponding to inner-products around a centroid, i...

We prove that every element of the polar cone to the closed convex cone of monotone transport maps can be represented as the divergence of a measure field taking values in the positive definite matrices.

The aim of this paper is to characterize in terms of scalar quasiconvexity the vector-valued functions which are K-quasiconvex with respect to a closed convex cone K in a Banach space. Our main result extends a wellknown characterization of K-quasiconvexity by means of extreme directions of the polar cone of K, obtained by Dinh The Luc in the particular case when K is a polyhedral cone generate...

Through the paper, X and Y are normed vector spaces; however, most of the results remain true in the more general setting of locally convex spaces. We denote by X∗ and Y∗ the topological dual spaces of X and Y . We consider a pointed closed convex cone Q ⊂ Y which introduces a partial order on Y by the equivalence y1 ≤Q y2 ⇔ y2 − y1 ∈Q; we also suppose, in general, that Q has nonempty interior....

Given a normed cone (X , p) and a subcone Y, we construct and study the quotient normed cone (X/Y, p̃) generated by Y . In particular we characterize the bicompleteness of (X/Y, p̃) in terms of the bicompleteness of (X , p), and prove that the dual quotient cone ((X/Y )∗,‖ · ‖p̃,u) can be identified as a distinguished subcone of the dual cone (X∗,‖ · ‖p,u). Furthermore, some parts of the theory ar...

The aim of this paper is to characterize in terms of classical (quasi)convexity of extended real-valued functions the set-valued maps which are K-(quasi)convex with respect to a convex cone K. In particular, we recover some known characterizations of K-(quasi)convex vector-valued functions, given by means of the polar cone of K.

Under suitable hypotheses on the function / , the two constrained minimization problems: MIN/ fyy subject to x > 0, -fy > 0; MAX/ fxx subject to y > 0, fx > 0; are well known each to be dual to the other. This symmetric duality result is now extended to a class of nonsmooth problems, assuming some convexity hypotheses. The first problem is generalized to: MINf(x,y) py subject to x e T,-p e S* n...