Convex spaces: Classification by differentiable convex functions

Approximations of differentiable convex functions on arbitrary convex polytopes

Let Xn := {xi}ni=0 be a given set of (n + 1) pairwise distinct points in R (called nodes or sample points), let P = conv(Xn), let f be a convex function with Lipschitz continuous gradient on P and λ := {λi}ni=0 be a set of barycentric coordinates with respect to the point set Xn. We analyze the error estimate between f and its barycentric approximation:

Characterizations of $L$-convex spaces

In this paper, the concepts of $L$-concave structures, concave $L$-interior operators and concave $L$-neighborhood systems are introduced. It is shown that the category of $L$-concave spaces and the category of concave $L$-interior spaces are isomorphic, and they are both isomorphic to the category of concave $L$-neighborhood systems whenever $L$ is a completely distributive lattice. Also, it i...

M-FUZZIFYING TOPOLOGICAL CONVEX SPACES

The main purpose of this paper is to introduce the compatibility of $M$-fuzzifying topologies and $M$-fuzzifying convexities, define an $M$-fuzzifying topological convex space, and give a method to generate an $M$-fuzzifying topological convex space. Some characterizations of $M$-fuzzifying topological convex spaces are presented. Finally, the notion of $M$-fuzzifying weak topologies is obtaine...

How to Define ”convex Functions” on Differentiable Manifolds

In the paper a class of families F(M) of functions defined on differentiable manifolds M with the following properties: 1F . if M is a linear manifold, then F(M) contains convex functions, 2F . F(·) is invariant under diffeomorphisms, 3F . each f ∈ F(M) is differentiable on a dense Gδ-set, is investigated.

Harmonic functions in non-locally convex spaces