Topologoies on Cones

Shedding Light On Cones

Daytime vision in vertebrates initiates with the absorption of light by cone photoreceptors (Rodieck, 1998), which generate signals for color discrimination (Sharpe et al., 1999). In humans, these cells are concentrated in a specialized part of the central retina called the fovea. This region of the eye operates over a wide range of intensities (Aguilar and Stiles, 1954) mediating high temporal...

Notes on Hyperbolicity Cones

minimize cx such that Ax = b and x ∈ Λ+, where c ∈ R, Ax = b is a system of linear equations and Λ+ is the closure of a so called hyperbolicity cone. Hyperbolic programming generalizes semidefinite programming, but it is not known to what extent since it is not known how general the hyperbolicity cones are. The rich algebraic structure of hyperbolicity cones makes hyperbolic programming an inte...

Notes on Asymptotic Cones

Let X be a metric space, {xi}i∈N a sequence of points in X (thought of as a sequence of centers), and {di}i∈N a sequence of scaling factors such that di → ∞, and an ultrafilter ω. I’m not going to get into the details of ultrafilters here, but the main property of an ultrafilter is that it lets you define the ultralimit limω ai of any bounded sequence ai. Ultralimits use the axiom of choice to ...

Some Results on Boundedness in Locally Convex Cones

On Polar Cones and Differentiability in Reflexive Banach Spaces

Let $X$ be a  Banach  space, $Csubset X$  be  a  closed  convex  set  included  in  a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a  bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set  $C$,  so that ${mathop{rm int}}(mathrm{dom} sigma_C) neqem...