Timeline for Linearization of cones
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 1, 2013 at 23:17 | comment | added | JHM |
@Brian: what do you mean by symmetric'? Because $0$-symmetric (i.e. $x\in K$ whenever $=x\in K$) means: every $0$-symmetric convex cone is already a linear subspace. What exactly is this log' function?
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| Jun 30, 2013 at 16:05 | comment | added | Felix Goldberg | @BrianLins Can you give me a reference where it's spelt out in some detail? Thanks a mil! | |
| Jun 30, 2013 at 14:06 | comment | added | Brian Lins | On a symmetric cone there is a logarithm function, defined on the interior of the cone which maps the cone to a subspace. For example, on the cone of real positive definite matrices, the range of the logarithm is all symmetric matrices. | |
| Jun 28, 2013 at 18:39 | comment | added | Felix Goldberg | A set $K$ is a convex cone if $\lambda x+\mu y \in K$ for $x,y \in K$ and $\lambda,\mu \in \mathbb{R}^{+}$. | |
| Jun 28, 2013 at 18:29 | comment | added | JHM | Could you please clarify your sense of `cone'. If $K$ is a cone in the sense that $x\in K$ implies $\mathbb{R} x \subset K$, then it appears that a closed convex cone is necessarily a linear subspace. | |
| Jun 28, 2013 at 7:00 | comment | added | Felix Goldberg | @J.Martel Farkas's lemma seems to me to be a sort of philosophical assignment; I am more interested in what you called case (ii) - any ideas will be greatly appreciated! | |
| Jun 28, 2013 at 6:49 | comment | added | JHM | We should possibly clarify whether (i) we are interested in a `philosophical' assignment of closed convex cones $K$ to linear subspaces or (ii) a fixed function $f$ such that the images $f(K)$ are always linear subspaces. In case (ii) I wonder whether or not the assumption that $f$ be continuous and maps linear subspaces (themselves closed convex cones) in $\mathbb{R}^n$ to linear subspaces in $\mathbb{R}^m$ necessarily forces $f$ to be, say, linear. | |
| Jun 27, 2013 at 23:01 | comment | added | Felix Goldberg | @SergeiIvanov I am afraid I have no example - that's why I am asking for existence. My motivation for this is to try to obtain results on cones from results on subspaces via this embedding. | |
| Jun 27, 2013 at 22:25 | comment | added | Sergei Ivanov | Can you clarify, perhaps by example, what you have in mind? Of course there is a linear map to $R^1$ such that $f(K)$ is either 0 or the entire line. | |
| Jun 27, 2013 at 21:18 | history | asked | Felix Goldberg | CC BY-SA 3.0 |