I would like to draw your attention to the 2002 survey by Volker Kaibel and Marc Pfetsch,
"Some Algorithmic Problems in Polytope Theory," [arXiv:math/0202204v1][1], which contains this on p.6:
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![KP Problems][2]
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As you probably know, an $\cal{H}$-description is by halfspaces, whereas a $\cal{V}$-description is by vertices.
Reference [20] is: R. M. Freund and J. B. Orlin, "On the complexity of four polyhedral set
containment problems," *Math. Program.*, 33 (1985), pp. 139–145.


  [1]: http://arxiv.org/abs/math/0202204
  [2]: http://cs.smith.edu/~orourke/MathOverflow/KPProblems.jpg