On curvatures of homogeneous convex cones

Characterization of the barrier parameter of homogeneous convex cones

We characterize the barrier parameter of the optimal self{concordant barriers for homogeneous cones. In particular, we prove that for homogeneous convex cones this parameter is the same as the rank of the corresponding Siegel domain. We also provide lower bounds on the barrier parameter in terms of the Carath eodory number of the cone. The bounds are tight for homogeneous self-dual cones.

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.

Some Results on Boundedness in Locally Convex Cones

Convex hypersurfaces of prescribed curvatures

For a smooth strictly convex closed hypersurface Σ in R, the Gauss map n : Σ → S is a diffeomorphism. A fundamental question in classical differential geometry concerns how much one can recover through the inverse Gauss map when some information is prescribed on S ([27]). This question has attracted much attention for more than a hundred years. The most notable example is probably the Minkowski...

Simplicial arrangements on convex cones

We introduce the notion of a Tits arrangement on a convex open cone as a special case of (infinite) simplicial arrangements. Such an object carries a simplicial structure similar to the geometric representation of Coxeter groups. The standard constructions of subarrangements and restrictions, which are known in the case of finite hyperplane arrangements, work as well in this more general settin...