Relative Schur-convexity on global NPC spaces

Journal of Mathematical Analysis and Applications, 2009

We prove an extension of Choquet's theorem to the framework of compact metric spaces with a global nonpositive curvature. Together with Sturm's extension [K.T. Sturm, Probability measures on metric spaces of nonpositive curvature, in: Pascal Auscher, et al.

Journal of Mathematical Analysis and Applications, 2008

The Knaster-Kuratowski-Mazurkiewicz covering theorem (KKM), is the basic ingredient in the proofs of many so-called "intersection" theorems and related fixed point theorems (including the famous Brouwer fixed point theorem). The KKM theorem was extended from R n to Hausdorff linear spaces by Ky Fan. There has subsequently been a plethora of attempts at extending the KKM type results to arbitrary topological spaces. Virtually all these involve the introduction of some sort of abstract convexity structure for a topological space, among others we could mention H-spaces and G-spaces. We have introduced a new abstract convexity structure that generalizes the concept of a metric space with a convex structure, introduced by E. Michael in [E. Michael, Convex structures and continuous selections, Canad. J. Math. 11 (1959) 556-575] and called a topological space endowed with this structure an M-space. In an article by Shie Park and Hoonjoo Kim [S. Park, H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173-187], the concepts of G-spaces and metric spaces with Michael's convex structure, were mentioned together but no kind of relationship was shown. In this article, we prove that G-spaces and M-spaces are close related. We also introduce here the concept of an L-space, which is inspired in the MC-spaces of J.V. Llinares [J.V. Llinares, Unified treatment of the problem of existence of maximal elements in binary relations: A characterization, J. Math. Econom. 29 (1998) 285-302], and establish relationships between the convexities of these spaces with the spaces previously mentioned.

2004

SIAM Journal on Optimization, 2016

Using techniques of convex analysis, we provide a direct proof of a recent characterization of convexity given in the setting of Banach spaces in [J. Saint Raymond, J. Nonlinear Convex Anal., 14 (2013), pp. 253-262]. Our results also extend this characterization to locally convex spaces under weaker conditions.

We introduce an extension of the standard Local-to-Global Principle used in the proof of the convexity theorems for the momentum map to handle closed maps that take values in a length metric space. As an application, this extension is used to study the convexity properties of the cylinder valued momentum map introduced by Condevaux, Dazord, and Molino. Mathematics Subject Index 2000: 53C23, 53D20. algebra action on a symplectic manifold and that generically takes values in an Abelian group isomorphic to a cylinder. The cylinder valued momentum map was introduced in [9] and carefully studied in in the context of reduction. Additionally, its local properties are as well understood as those for the standard momentum map . The generalized Local-to-Global Principle allows us to extend to the cylinder valued momentum map the knowledge that we have about the convexity properties of the classical momentum map. The metric approach seems to be the best adapted generalization of the classical setup to our problem since, under certain hypotheses related to the topological nature of the Hamiltonian holonomy of the problem (a concept defined carefully later on), the target space of the cylinder valued momentum map has an associated canonical length space structure.

Kragujevac journal of mathematics, 2022

The aim of this work is to investigate the Schur convexity, Schur geometrically convexity, Schur harmonically convexity and Schur power convexity of some special functions. Some sufficient conditions are obtained to guarantee the above-mentioned properties satisfy. We attain some special inequalities. Also, we obtain some applications of main results.

Metroeconomica, 1968

In this note, we discuss the definition of the S1-convexity Phenomenon. We first make use of some results we have attained for KS2 in the past, such as those contained in [1], to refine the definition of the phenomenon. We then observe that easy counter-examples to the claim K1 extends K0 are found. Finally, we make use of one theorem from [2] and a new theorem that appears to be a supplement to that one to infer that does not properly extend K0 in both its original and its revised version.

2017

Definition 1.3 (Global NPC/CAT(0)) Suppose T is a geodesic space with the metric d. Consider a triangle T in T of side lengths a, b, c, and build a comparison triangle T ′ with the same lengths in Euclidean plane. Consider a chord of length l in T which connects two points on the boundary of T ; there is a corresponding comparison chord in T ′, say of length l′. If for every triangle T in T and every chord in T we have l ≤ l′, T is said to be global nonpositively curved (NPC) or CAT(0).