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The Voronoi diagram is a geometric object which is widely used in many areas.Recently it has been shown that under mild conditions Voronoi diagrams have acertain continuity property: small perturbations of the sites yield smallperturbations in the shapes of the corresponding Voronoi cells. However, thisresult is based on the assumption that the ambient normed space is uniformlyconvex. Unfortunately, simple counterexamples show that if uniform convexity isremoved, then instability can occur. Since Voronoi diagrams in normed spaceswhich are not uniformly convex do appear in theory and practice, e.g., in theplane with the Manhattan (ell_1) distance, it is natural to ask whether thestability property can be generalized to them, perhaps under additionalassumptions. This paper shows that this is indeed the case assuming the unitsphere of the space has a certain (non-exotic) structure and the sites satisfya certain "general position" condition related to it. The condition on the unitsphere is that it can be decomposed into at most one "rotund part" and at mostfinitely many non-degenerate convex parts. Along the way certain topologicalproperties of Votonoi cells (e.g., that the induced bisectors are not "fat")are proved.

The geometric stability of Voronoi diagrams in normed spaces which are not uniformly convex