Let $P\subset \Bbb R^d$ be a convex polytope. Suppose that I know
- its combinatorial type (aka. the face-lattice),
- the length $\ell_i$ of each edge, and
- the distance $r_i$ of each vertex from the origin.
Question: Does this already determine $P$ (up to orthogonal transformation)?
This is the case if all $\ell_i$ are the same, and all $r_i$ are the same (see this question). But what if they are not the same? What if I do not know the combinatorial type but only the edge-graph?
I am not sure whether the formulation of my question was too vague, so below I added a second equivalent version of what I am asking:
Given two combinatorailly equivalent polytopes $P_1,P_2\subset\Bbb R^d$, and a corresponding face-lattice isomorphism $\phi:\mathcal F(P_1)\to\mathcal F(P_2)$. Now suppose that each edge $e\in\mathcal F_1(P_2)$ has the same length as $\phi(e)\in\mathcal F_1(P_2)$, and that each vertex $v\in\mathcal F_0(P_1)$ has the same distance from the origin as $\phi(v)\in\mathcal F_0(P_2)$. Is it then true that $P_1$ and $P_2$ are congruent (related by an orthogonal transformation)?