Consider the family of convex simplicial polytopes with vertices in the unit sphere of $\mathbb{R}^n$ which have the origin as an interior point.

My question is the following:

Let $P, P'$ be two non-congruent combinatorially identical polytopes from the above family, with vertices of each polytope labelled so that the face lattices of the two polytopes are identical. Is it possible that $\|a-b\|\leq\|a'-b'\|$ whenever edge $\{a,b\}$ in $P$ corresponds to edge $\{a',b'\}$ in $P'$?

In other words,

can you perturb one such polytope to another making all the edges grow?

This is impossible in $\mathbb{R}^2$: Take an inscribed polygon from the family and perturb it so that it remains in the family and has the same combinatorial structure. If all the edges grow, then all the central angles grow, which would make those central angles add up to a value larger than $2\pi$. Therefore in a perturbed polygon, if some edges grow then other edges must contract.

Does this idea transfer to dimension $n=3$, or more generally to $n\geq3$? And, if yes, is there a reference?

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