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Li Yutong
Li Yutong
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Let $P \subset \mathbb R^n$ be a compact rational polytope (the affine space spanned by $P$ may not be $\mathbb R^n$). Let $G \subset {\bf GL}(n,\mathbb Z)$ be an arithmetic group. Let $$C = {\rm Conv}(\bigcup_{g\in G} \ g\cdot P)$$ be the convex hull.

Question: does there exist a compact rational polytope $Q \subset C$ such that $\bigcup_{g\in G}\ g\cdot G = C$$\bigcup_{g\in G}\ g\cdot Q = C$ ?

Let $P \subset \mathbb R^n$ be a compact rational polytope (the affine space spanned by $P$ may not be $\mathbb R^n$). Let $G \subset {\bf GL}(n,\mathbb Z)$ be an arithmetic group. Let $$C = {\rm Conv}(\bigcup_{g\in G} \ g\cdot P)$$ be the convex hull.

Question: does there exist a compact rational polytope $Q \subset C$ such that $\bigcup_{g\in G}\ g\cdot G = C$ ?

Let $P \subset \mathbb R^n$ be a compact rational polytope (the affine space spanned by $P$ may not be $\mathbb R^n$). Let $G \subset {\bf GL}(n,\mathbb Z)$ be an arithmetic group. Let $$C = {\rm Conv}(\bigcup_{g\in G} \ g\cdot P)$$ be the convex hull.

Question: does there exist a compact rational polytope $Q \subset C$ such that $\bigcup_{g\in G}\ g\cdot Q = C$ ?

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Li Yutong
Li Yutong
  • 3.5k
  • 16
  • 34

Existence of a fundamental domain for the convex hull of group action on a rational polytope

Let $P \subset \mathbb R^n$ be a compact rational polytope (the affine space spanned by $P$ may not be $\mathbb R^n$). Let $G \subset {\bf GL}(n,\mathbb Z)$ be an arithmetic group. Let $$C = {\rm Conv}(\bigcup_{g\in G} \ g\cdot P)$$ be the convex hull.

Question: does there exist a compact rational polytope $Q \subset C$ such that $\bigcup_{g\in G}\ g\cdot G = C$ ?