I'm considering the dual cone $K^*$ of a non-convex cone $K$. I came up with a theory that $K^{**}$ is the closure of convex hull of $K$. Then I wonder whether $(K^*)^{**}=(K^{**})^*$ holds for any cone $K$. If it's true, then the dual cone of a non-convex cone $K$ will be the same as the dual cone of its closure of convex hull.
$\begingroup$
$\endgroup$
4
-
3$\begingroup$ Perhaps you might tell us what $K^*$ and $K^{**}$ mean? $\endgroup$– Gerry MyersonFeb 22, 2022 at 4:01
-
1$\begingroup$ Does "any" mean "some" or "every"? $\endgroup$– Jukka KohonenFeb 22, 2022 at 7:02
-
2$\begingroup$ The statement, that $K^{**}$ is the closed convex hull of $K$ would make sense for a cone in a reflective real Banach space $V$. In this case, a cone would be a subset $K$, which is closed under multiplication by positive scalars. Then $K'\subset V'$ is defined as the set of all $f\in K'$ with $f(K)\ge 0$ and the statement would be that $K''=(K')'$ is the closed convex hull of $\delta(K)$, where $\delta:V\to V''$ is the isomorphism given by evaluation. If that is the question, then the answer is yes by Hahn-Banach. $\endgroup$– user473423Feb 22, 2022 at 7:52
-
1$\begingroup$ @GerryMyerson The dual cone: en.wikipedia.org/wiki/Dual_cone_and_polar_cone $\endgroup$– DirkFeb 22, 2022 at 8:54
Add a comment
|