"All vertices of a finite n-dimensional isogonal figure exist on an (n-1)-sphere." — Wikipedia.
I've been investigating on the subject of regular polytopes recently, and I've found about the subject of Archimedean or vertex-transitive polytopes recently. A characteristic I have seen all of them have in common, and has been commonly cited to be true (but not proven) is the fact that they all seem to be able to be circumscribed in a sphere of a dimension lower (of or the same dimension, considering the interior) than the dimension they are in. That is, every vertex-transitive polygon is cyclic (this turns out to be quite easy to prove), every vertex-transitive polyhedron can be inscribed in a sphere...
I've been trying to prove this but I'm simply stuck. I managed to show that a vertex-transitive polyhedron can be inscribed in a sphere iff any of its vertices and the faces around it can. (The idea is to simply consider to adjacent faces and the two vertices they belong, and assume both can be circumscribed). But afterwards, it seems to be quite hard to make any more statements that can help.
Another thing I considered was using the definition of "vertex-transitive" itself, but I'm not sure how symmetries and so could actually help.
And finally I even thought about Wythoff construction and its generalizations, but even if such an approach worked, the non-Wythoffian figures would have to be considered separately, and since there are an unknown number of them (especially when you don't even require regular faces), this looks like it comes to a dead end.
Any help is appreciated.