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I have two cones $A$ and $B$ in a Euclidean space. I want to show that $\mathrm{Conv}\,A=\mathrm{Dual}\,B$; that is, $a\in\mathrm{Conv}\,A$ if and only if $\langle a,b\rangle\ge 0$ for any $b\in B$.

The inclusion $\mathrm{Conv}\,A\subset \mathrm{Dual}\, B$ (or equivalently $\mathrm{Dual}\,A\supset \mathrm{Conv}\,B$) is easy --- one has to check $\langle a,b\rangle\ge 0$ for pairs $a\in A$ and $b\in B$. It remains to show the opposite inclusion $\mathrm{Conv}\,A\supset \mathrm{Dual}\,B$. I see that in principle it can be done by calculations.

Do you know tricks that help to prove the opposite inclusion $\mathrm{Conv}\,A\supset \mathrm{Dual}\,B$? Is there software that can help?

The cones $A$ and $B$ are very concrete; they are given by few algebraic identities and inequalities in $\mathbb R^{10}$. (In principle I could describe the sets of vectors here, but I do not think it might help.)

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    $\begingroup$ I assume that $\operatorname{Conv} A$ denotes the convex hull of $A$? But why do you need to take the convex hull of a cone, given that a cone is already convex? $\endgroup$
    – Jochen Glueck
    Commented Jun 19, 2020 at 21:27
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    $\begingroup$ @JochenGlueck I did not say that cones are convex (in fact they are not). $\endgroup$
    – Anton Petrunin
    Commented Jun 20, 2020 at 3:51
  • $\begingroup$ Thank you very much for your response! Hmm, may I suggest to include a definition of your usage of the word cone in the question? (I just browsed half a dozen books about ordered vector spaces; many of them define a cone indeed to be always convex, while some others do not - and there also seem to be some further differences between the various definitions of a cone in the literature). $\endgroup$
    – Jochen Glueck
    Commented Jun 20, 2020 at 12:49
  • $\begingroup$ @JochenGlueck a set $X$ is a cone if $v\in X$ implies that $\lambda{\cdot}v\in X$ for any $\lambda\ge0$. $\endgroup$
    – Anton Petrunin
    Commented Jun 20, 2020 at 19:08
  • $\begingroup$ Thank you very much. $\endgroup$
    – Jochen Glueck
    Commented Jun 20, 2020 at 20:04

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