Regularity of locally convex surfaces

Boundary Torsion and Convex Caps of Locally Convex Surfaces

We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least 4 times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the 4 vertex theorem of Sedykh for convex space curves, and thus constitutes a far r...

Bornological Completion of Locally Convex Cones

In this paper, firstly, we obtain some new results about bornological convergence in locally convex cones (which was studied in [1]) and then we introduce the concept of bornological completion for locally convex cones. Also, we prove that the completion of a bornological locally convex cone is bornological. We illustrate the main result by an example.

Surfaces with Degenerate Cr Singularities That Are Locally Polynomially Convex

A compact subset K ⊂ C is said to be polynomially convex if for every point ζ / ∈ K, there exists a holomorphic polynomial P such that P (ζ) = 1 and supK |P | < 1. K is said to be locally polynomially convex at a point p ∈ K if there exists a closed ball B(p) centered at p such that K ∩ B(p) is polynomially convex. In general, it is difficult to determine whether a given compact K ⊂ C is polyno...

On the dual of certain locally convex function spaces

In this paper, we first introduce some function spaces, with certain locally convex topologies, closely  related to the space of real-valued continuous functions on $X$,  where $X$ is a $C$-distinguished topological space. Then, we show that their dual spaces can be identified in a natural way with certain spaces of Radon measures.

Some Results on Boundedness in Locally Convex Cones