Given a class $${{\mathcal {P}}}$$ of Banach spaces, locally convex space (LCS) E is called multi- if can be isomorphically embedded into product spaces that belong to . We investigate the question whether free L(X) strongly nuclear, Schwartz, multi-Hilbert or multi-reflexive. If X Tychonoff containing an infinite compact subset then, as it follows from results Außenhofer (Topol Appl 134:90–102...

As topology-like mathematical structures, convex structures can be characterized by betweenness relations via (restricted) hull operators in spaces. In a topological approach, the aim of this paper is to present fuzzy counterpart based on points Concretely, notion L-betweenness restricted L-hull introduced. Firstly, it proved that are categorically isomorphic and L-remotehood systems, respectiv...

‎Let $X$ be a real normed  space, then  $C(subseteq X)$  is  functionally  convex  (briefly, $F$-convex), if  $T(C)subseteq Bbb R $ is  convex for all bounded linear transformations $Tin B(X,R)$; and $K(subseteq X)$  is  functionally   closed (briefly, $F$-closed), if  $T(K)subseteq Bbb R $ is  closed  for all bounded linear transformations $Tin B(X,R)$. We improve the    Krein-Milman theorem  ...

A game on a convex geometry is a real-valued function defined on the family L of the closed sets of a closure operator which satisfies the finite Minkowski-KreinMilman property. If L is the Boolean algebra 2 then we obtain an n-person cooperative game. We will extend the work of Weber on probabilistic values to games on convex geometries. As a result, we obtain a family of axioms that give rise...

Let L = ∆+Z for a C1 vector field Z on a complete Riemannian manifold possibly with a boundary. By using the uniform distance, a number of transportation-cost inequalities on the path space for the (reflecting) L-diffusion process are proved to be equivalent to the curvature condition Ric−∇Z ≥ −K and the convexity of the boundary (if exists). These inequalities are new even for manifolds withou...

We consider a different L-Minkowski combination of compact sets in R than the one introduced by Firey and we prove an L-BrunnMinkowski inequality, p ∈ [0, 1], for a general class of measures called convex measures that includes log-concave measures, under unconditional assumptions. As a consequence, we derive concavity properties of the function t 7→ μ(t 1 pA), p ∈ (0, 1], for unconditional con...

Let K and L be compact convex sets in Rn. The following two statements are shown to be equivalent: (i) For every polytope Q ⊆ K having at most n+ 1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 ≤ d ≤ n − 1. It is also shown that the following two statements are equivalent: (i) For every polytope Q ⊆ K having at most d+ 1 vertices, L contains a translate of Q. (i...

In the classical model of cooperative games it is generally assumed that there are no restrictions on cooperation and hence, every subset of players is a feasible coalition. However, in many social and economic situations, this model does not apply. Examples are provided by local public goods which are supplied by local communities, social and sports clubs, labor unions, political parties, and ...