We propose a new abstract definition of equilibrium in the spirit of competitive equilibrium: a profile of alternatives and a public ordering (expressing prestige, price or a social norm), such that each agent prefers his assigned alternative to all lower-ranked ones. The equilibrium operates in an abstract setting built upon a concept of convexity borrowed from Convex Geometry. We apply the co...

We propose a new abstract definition of competitive equilibrium: a profile of alternatives and a public ordering (prestige, price or a social norm) such that each agent prefers his assigned alternative to all lower-ranked ones. The equilibrium operates in an abstract setting built upon a concept of convexity borrowed from Convex Geometry. The "magic" of linear equilibrium prices is put into per...

One of the application areas of abstract convexity is inequality theory. In this work, the authors seek to derive new inequalities by sharpening well-known inequalities by the use of abstract convexity. Cauchy-Schwarz inequality, Minkowski inequality and well-known mean inequalities are studied in this sense, concrete results are obtained for some of them. Mathematics subject classification (20...

Restricted-orientation convexity, also called O-convexity, is the study of geometric objects whose intersection with lines from some xed set is empty or connected. We introduce restricted-orientation halfspaces, which are an O-convexity analog of halfspaces, explore their properties, and demonstrate their relationship to restricted-orientation convex sets. ABSTRACT Restricted-orientation convex...

Comparative convexity is a generalization of convexity relying on abstract notions of means. We define the (skew) Jensen divergence and the Jensen diversity from the viewpoint of comparative convexity, and show how to obtain the generalized Bregman divergences as limit cases of skewed Jensen divergences. In particular, we report explicit formula of these generalized Bregman divergences when con...

Correspondence: gaoyingimu@163. com Department of Mathematics, Chongqing Normal University, Chongqing 400047, China Abstract In this paper, a pair of nondifferentiable multiobjective fractional programming problems is formulated. For a differentiable function, we introduce the definition of higher-order (F, a, r, d)-convexity, which extends some kinds of generalized convexity, such as second or...

CONVEXITY AND CONE-VEXING ABSTRACTIONS 3 X × Φ and the cone ∆2(X) × {0} × R+ constitute a nonoblate pair in X × Y × R. Cones K1 and K2 in a topological vector space X are in general position provided that (1) the algebraic span of K1 and K2 is some subspace X0 ⊂ X; i.e., X0 = K1 −K2 = K2 −K1; (2) the subspace X0 is complemented; i.e., there exists a continuous projection P : X → X such that P (...