There is no such a characterization. The characterization of Voronoi cells is known in dimensions at most 5, respectively. You can choose from the full list those ones, which have a circumscribed ball. (I don't know that there is or not such a complete list, but I think that probably it is not too long...) In higher dimensions the complete lists of Dirichlet-Voronoi cells is unknown, respectively. It semms to be interesting to find a method which gives the cells with circumscribed ball.
Dear Andras, I would have thought that if you have a polytope that is both space-filling and spherical that would be a very strong constraint.....I am not sure what you mean by list; already in dimension 3 there is a parametrization not really a discrete list....more like a variety...
Dear Patrick, In a fixed dimension there are finitely many combinatorial possibility to get a Voronoi polytope of a lattice. More generally the combinatorial types of those polytopes which translates fill the space (these are parallelotopes or extremal bodies)
known in dimensions at most 5. In such a class every two polytopes are affine equivalent, and it is possible that (you are right and) there is a variety of non-congruant Voronoi cells,but in the general case the situation is that the class contains preciselly one Voronoi cell (without similarity). (The orthogonality of the relevant vectors to its facets is a very strong assumption, see Voronoi conjecture...) You also add a strong assumption by the existance of the circumscribe ball. So I think that for a fixed dimension there is a finite combinatorial list and possible that some elements of the list contains a variety of non-congruant poltopes .
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