Let $P\subset \Bbb R^n$ be an inscribed convex polytope, that is, all its vertices are on a common sphere of radius $r$.
Let $G$ be the edge-graph of $P$. For convenience, assume $V(G)=\{1,\dotsc,s\}$. Let $\ell\_{ij}$ denote the length of the edge of $P$ corresponding to $ij\in E(G)$.



> 
> **Question.** Let $p\_1,\dotsc,p\_s\in\Bbb R^{m}$ be points so that
> 
> 
> * the points are on a common sphere $S$,
> * $\lVert p\_i-p\_j\rVert\le\ell\_{ij}$ for all $ij\in E(G)$,
> * the center of $S$ lies in the convex hull $\operatorname{conv}\{p\_1,\dotsc,p\_s\}$.
> 
> 
> Is it then true that the radius of $S$ is at most $r$? If no, does this change if $n=m$?
> 
> 
> 


In other words, are the skeleta of inscribed polytopes "as expanded as possible" for the given edge-lengths?


Note that the condition on the convex hull is necessary. Without this we could choose an arbitrarily large sphere $S$, and place all the $p\_1,\dotsc,p\_s$ in an arbitrarily small patch of $S$, so that $S$ is their circumsphere.




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**The case $n=m=2$**


I will demonstrate my general ideal on the case $n=m=2$, which I hope to somehow generalize to all cases with $n=m$. I do not yet have an idea for $m>n$.


Let $P\subset\Bbb R^2$ be an inscribed polygon with circumradius $r$ and vertices $v\_1,...,v\_s$ in circular order. Let $\alpha\_i$ be the angle between $v\_i$ and $v\_{i+1}$ (indices mod $s$) as seen from the circumcenter. Then $\alpha\_1+\dots+\alpha\_s=2\pi$.


Suppose now that we have such a set of points $p\_1,...,p\_s$ with circumradius $r'>r$. Let $\beta\_i$ be the angle between $p\_i$ and $p\_{i+1}$ as seen from the circumcenter. Since $\|p\_i-p\_{i+1}\|\le \|v\_i-v\_{i+1}\|$ but also $\|p\_i\|>\|v\_i\|$ it is easy to see that $\beta\_i<\alpha\_i$. In particular, $\beta\_1+\cdots+\beta\_s<2\pi$, and the closed polyline with vertices $p\_1,...,p\_s$ must have zero winding number around the circumcenter. But since the convex hull of the $p\_i$ contains the origin, there are three points $p\_{i\_1},p\_{i\_2},p\_{i\_3}$ with $i\_1<i\_2<i\_3$ so that already the convex hull of these contains the circumcenter. It is then easy to see that $$\beta\_{i\_1}+\cdots+\beta\_{i\_2}+\cdots+\beta\_{i\_3}\ge \pi$$
and
$$\beta\_{i\_3+1}+\beta\_{i\_3+2}+\cdots+\beta\_{i\_1-1}\ge \pi,$$
in contradiction to $\beta\_1+\cdots+\beta\_s<2\pi$. $\;\square$


To generalize this, one would need to find a suitable notion of "winding number" (mapping degree) for point arrangements with $n>3$, somehow using the face structure provided by the polytope. One might then be able to construct a similar argument using space angles (aka fractions of the sphere, as described by Matt F. in the comments).



Given the skeleton of an inscribed polytope. If I move the vertices so that no edge increases in length, can the circumradius still get larger?
The answer to the question is unfortunately no, in the case of $n=m=3$. There is a simple example to illustrate this.


Let $P$ be the cube with vertices $(\pm \frac{1}{2},\pm \frac{1}{2},\pm \frac{1}{2})$. Very obviously, every edge is of length $1$, and a quick calculation shows this is inscribed upon a sphere of radius $\frac{\sqrt{3}}{2}$.


We now describe the locations of 8 vertices, one for each sign-string of length 3. Vertex $V\_{sign}$ with $s=\#$positive entries of $sign$ is at the point $(sin(\frac{s\pi}{3}),cos(\frac{s\pi}{3}),0)$. It is easy to check that the lengths of the edges is exactly preserved. However, these vertices lie on a sphere of radius $1 > \frac{\sqrt{3}}{2}$.


There are three things wrong with this example, two of which are immediately fixable. First, $\vec{0}$ is not interior. A small reduction of the radius can fix this. Secondly, the convex hull of these points is not full dimensional. Again a small reduction of the radius combined with some small perturbation in the third coordinate fixes this.


Finally, the images of the faces of $P$ form a complex which does not contain the origin. This is both what allows the example to exist, and makes it a not satisfying example. Perhaps an additional condition which could contribute to a winding number argument could be: the cone generated by all edges incident to a vertex contains the origin. This could replace that the origin is in the convex hull of the points.


