Is there a non-orthogonal linear deformation of a polytope that preserves edge-lengths and vertex-origin-distances?

Question

Is there a polytope $P\subset\Bbb R^d$ (convex hull of finitely many points, not contained in a proper affine subspace), and a linear, but non-orthogonal transformation $T\in\mathrm{GL}(\Bbb R^d)\setminus\mathrm O(\Bbb R^d)$, so that

• $T$ preserves all the edge lengths of $P$, and
• $T$ preserves the distance of every vertex of $P$ from the origin?

If I require only one of these, then the answer is Yes, as demonstrated in the following images:

I know that the answer is No if the polytope has a single neighborly facet (e.g. a simplex), but I have no idea for the general case.

Answer 1

The answer is No, and we just need $\mathrm{lin}(P)=\Bbb R^d$ rather than $\mathrm{aff}(P)=\Bbb R^d$.

Proof.

Note that a linear map $T$ preserving edge-lengths and vertex-origin-distances can be equivalently expressed as $\def\<{\langle}\def\>{\rangle}\<Tv,Tw\>=\langle v,w\>$ for vertices $v,w\in\mathcal F_0(P)$, whenever $v=w$ or $v$ and $w$ are adjacent.

On the other hand, if $\mathrm{lin}(P)=\Bbb R^d$, then $T$ being orthogonal is the same as $\<Tv,Tw\>=\<v,w\>$ for all vertices $v,w\in\mathcal F_0(P)$. We prove that this follows from the weaker statement above.

For this, choose arbitrary $v,w\in\mathcal F_0(P)$. It is well known that $w$ is contained in the cone $v+\mathrm{cone}\{u-v\mid \text{$u$ is a neighbor of $v$}\}$. That is, there are neighbors $u_1,...,u_k\in\mathcal F_0(P)$ of $v$ so that

$$w=v+\alpha_1 (u_1-v) + \cdots + \alpha_k (u_k-v) = \beta_0 v + \beta_1 u_1 + \cdots + \beta_k u_k.$$

But then we can compute

$$\<v,w\> = \beta_0\<v,v\> + \beta_1\<v,u_1\> + \cdots + \beta_k \<v,u_k\>,$$

and since all inner product on the right are preserved by $T$, so is the inner product on the left, and we are done.

$\square$

Answer 2

This general topic is explored in detail in https://arxiv.org/abs/1605.07911 (I am a co-author). For your particular question, see Lemma 4.10 and and Prop 3.4.