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

_{(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.

_{(Wikipedia image: 600-cell)}