Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?
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asked Jun 27, 2013 at 21:18
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$\begingroup$ 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. $\endgroup$– Sergei IvanovCommented Jun 27, 2013 at 22:25
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$\begingroup$ @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. $\endgroup$– Felix GoldbergCommented Jun 27, 2013 at 23:01
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$\begingroup$ 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. $\endgroup$– JHMCommented Jun 28, 2013 at 6:49
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$\begingroup$ @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! $\endgroup$– Felix GoldbergCommented Jun 28, 2013 at 7:00
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$\begingroup$ 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. $\endgroup$– JHMCommented Jun 28, 2013 at 18:29
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