In a regular Voronoi diagram, the distance function from seed points grows as circles or spheres of identical sizes from seed points. In weighted Voronoi, the growing spheres have different sizes because they grow at different rates.
Consider another variation where instead of spheres, the distance functions grow as ellipses or ellipsoids. In other words, the distance function is sqrt(xTMix) for positive definite Mi. Each seed point i can have a different M matrix, so the elliposoids can have different orientations and growth rates.
Inspired how crystals prefer to grow in certain directions faster than others, probably corresponding to crystal structure. Eventually space will be filled by a bunch of crystal domains, with boundaries between them. We could add the further constraint of crystal weighted Voronoi, requiring connectedness of regions to the seed point.
Inspired by Widmanstatten patterns. One can take a 2D cross-section of any higher dimensional Voronoi diagram. Do the cross sections look different depending on what dimension they came from?
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