Some physicists suggest that to more adequately describe the causal structure of space-time, it is necessary to go beyond the usual pseudoRiemannian causality, to a more general Finsler causality. In this general case, the set of all the events which can be influenced by a given event is, locally, a generic convex cone, and not necessarily a pseudoReimannian-style quadratic cone. Since all curr...

In this paper, a local approach to the concept of Hudetz $g$-entropy is presented. The introduced concept is stated in terms of Hudetz $g$-entropy. This representation is based on the concept of $g$-ergodic decomposition which is a result of the Choquet's representation Theorem for compact convex metrizable subsets of locally convex spaces.

We introduce a new class of adaptive methods for optimization problems posed on the cone of convex functions. Among the various mathematical problems which posses such a formulation, the Monopolist problem [24, 10] arising in economics is our main motivation. Consider a two dimensional domain Ω, sampled on a grid X of N points. We show that the cone Conv(X) of restrictions to X of convex functi...

Let F be a fully nonlinear second-order partial differential subequation of degenerate elliptic type on a manifold X. We study the question: Which closed subsets E ⊂ X have the property that every F -subharmonic function (subsolution) on X−E, which is locally bounded across E, extends to an F -subharmonic function on X. We also study the related question for F -harmonic functions (solutions) wh...

Let F be a fully nonlinear second-order partial differential subequation of degenerate elliptic type on a manifold X. We study the question: Which closed subsets E ⊂ X have the property that every F -subharmonic function (subsolution) on X − E, which is locally bounded across E, extends to an F -subharmonic function on X. We also study the related question for F harmonic functions (solutions) w...

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....

We give some new regularity conditions for Fenchel duality in separated locally convex vector spaces, written in terms of the notion of quasi interior and quasi-relative interior, respectively. We provide also an example of a convex optimization problem for which the classical generalized interior-point conditions given so far in the literature cannot be applied, while the one given by us is ap...

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in H satisfying f(κ) = σ ∈ (0, 1) with a prescribed asymptotic boundary Γ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if Γ is (Euclidean) starshaped, the solution is uni...

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in H n+1 satisfying f(κ) = σ ∈ (0, 1) with a prescribed asymptotic boundary Γ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if Γ is (Euclidean) starshaped, the solution is...