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Q1. Are there convex polytopes combinatorially equivalent to each of the regular polytopes that are realized with integer vertex coordinates?


           dodecaplex
           (Wikipedia image: 120-cell)

There are six regular polytopes in $\mathbb{R}^4$ (above is the 120-cell) and three in higher dimensions.

Q2. Are the realization spaces of convex polytopes combinatorially equivalent to each of the regular polytopes open balls topologically?

The answer to both of these questions is Yes in $\mathbb{R}^3$, but without the regularity stipulation, the answers are No already in four dimensions: there are polytopes only realizable with irrational coordinates, and polytopes with (highly) disconnected realization spaces. See Jürgen Richter-Gebert, Günter M. Ziegler, "Realization spaces of 4-polytopes are universal." Bulletin of the AMS, Volume 32, Number 4, October 1995, Pages 403-412. arXiv link.

I believe the answer to Q1 is Yes, but I am much less certain of Q2. Perhaps those familiar with Richter-Gebert's proof technique and the surrounding literature (e.g., Mnev’s Universality Theorem) can see through this easily. I'd appreciate help—Thanks!

Update. Q1 has been answered positively by Igor Pak. But to show this is not obvious, it is apparently open whether the "first truncation" of the 600-cell is rational: Günter Ziegler, "Non-rational configurations, polytopes, and surfaces," arXiv link. On Q2, a colleague pointed me to a theorem that appears on p.92 in Grünbaum's Convex Polytopes book, which I cannot access at the moment. But Ziegler quotes a theorem from Grünbaum that proves that the realization space of any convex polytope is a semialgebraic set. Which of course does not resolve Q2.
          600-cell
           (Wikipedia image: 600-cell)

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2 Answers

Well, all simplicial polytopes can be made rational (i.e. to have rational coordinates) by perturbing vertices. Similarly, all simple polytopes can be made rational by perturbing hyperplanes. This covers all regular polytopes except for the 24-cell which already has a natural rational realization (see WP page). This resolves Q1. I am not sure about Q2.

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Nice, clean argument, Igor---Thanks! –  Joseph O'Rourke Sep 14 '12 at 18:45
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Answering asymptotically 1/3 of the question: the realization space of the simplex is contractible: when parametrized by the squares of the lengths, the space of simplices is a convex cone (a linear transformation of the PSD cone). This, together with some related results, is discussed in this old preprint.

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"Some observations on the simplex." Thanks, Igor! (Is 2003 old?) –  Joseph O'Rourke Sep 15 '12 at 14:31
    
@Joseph: for an unpublished preprint, yes :( –  Igor Rivin Sep 15 '12 at 15:13
    
Igor, for simplex the realization space is $GL(n,R)/O(n)$, which is contractile. –  Misha Sep 15 '12 at 16:57
    
@Misha: I am guessing that proving that the homogeneous space is contractible is harder than the direct proof of the result for the simplex... –  Igor Rivin Sep 15 '12 at 17:39
    
@Misha: I am guessing that in this case the contractibility follows from the polar decomposition theorem, which is harder (but not incredibly much harder) than the fact that symmetric positive semidefinite matrices are similar to diagonal matrices, which is how the result is proved in that preprint. –  Igor Rivin Sep 15 '12 at 23:37
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