Let $C,C'$ be rational polyhedral cones in $\mathbb R^n$ both with non-empty interior. Rational means they are generated by vectors with rational entries. One says that $C,C'$ are *isomorphic* if there is a unimodular integer matrix $A\in Gl(n,\mathbb Z)$ such that $AC=C'$. 



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> > **Question.** What is the computational complexity of deciding whether $C,C'$ are isomorphic? I'm happy to consider either that the cones are given by finite generating sets (say with integer entries) or by defining inequalities (say over $\mathbb Z$). 
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Computational complexity of deciding isomorphism of rational polyhedral cones
According to the abstract of "[On the complexity of polytope isomorphism problems](http://arxiv.org/abs/math/0106093)", 



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> [W]e derive that the problems to decide whether two polytopes, given either by vertex or by facet descriptions, are projectively or affinely isomorphic, are graph isomorphism hard. 
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Skimming the proof of Theorem 3, I think it should also establish that your problem is GI-Hard.


This paper has [6 citations in MathSciNet](http://www.ams.org/mathscinet/search/publications.html?refcit=1996205&loc=refcit), none of which appear to address this issue, so it is possible that this is the state of the art.


