in reply to Re^3: Is it possible to get a 'hole' in a Veronoi diagram?
in thread Better maps with Math::Geometry::Voronoi, and a Challenge for Math Monks

I don't think you can get even a polygon that is not convex, because by construction, you start out with a convex polygon (the whole space) and all areas you're clipping away from that are using lines/half-spaces perpendicular to the line connecting the two points. If you assume a metric space with a symmetric metric respecting the triangle inequality, I have the feeling that you encounter a contradiction fairly quickly, but I haven't written down any formal proof either :)

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Re^5: Is it possible to get a 'hole' in a Veronoi diagram?
by BrowserUk (Patriarch) on Jul 03, 2008 at 19:50 UTC

    That all sound plausible to me. Greek, but plausible :)

    My 'proof' is somewhat erm, simpler. Working with Sam's phone box analogy from above, for there to be a hole in the middle of the phone boxes would imply an area of space between them that isn't "closest" to any of them. Which just doesn't make any sense.

    Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
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