# Is there any work in topological data analysis on something like “Voronoi complexes”?

Given a finite set $$X \subset \mathbb{R}^n$$, we can of course construct the corresponding Čech or Vietoris-Rips filtration. At each level of this filtration the scale parameter is fixed and unrelated to $$X$$. If $$X$$ has regions of variable density, which is more typical than not in practice, then this variability affects the resulting invariants. While this may be a feature and not a bug, I am curious to know if (and especially where) alternative filtrations similar to the following construction may have been studied.

Take the Voronoi regions corresponding to $$X$$ and dilate each of them (about their respective "anchor points") by a fixed scale, then build a simplicial complex using the intersections of these dilated regions.

For what it's worth, a casual search of Google Scholar yielded nothing of obvious relevance.

• Have you looked at so-called "alpha complexes"? I think they may be related, though I am not an expert. – Aurel Jan 29 '19 at 16:01
• @Aurel--alpha complexes certainly involve the Voronoi regions, but the scale of the underlying balls is fixed. – Steve Huntsman Jan 29 '19 at 16:24
• As Aurel writes, alpha complexes and weighted alpha complexes are related in spirit in the sense that they involve Voronoi regions. Similarly, you may be interested in checking out witness complexes, which are related in spirit in the sense that they involve density (points in varying regions of density are weighted differently). Other folks have considered nerve or clique complexes of ellipsoidal shapes where the ellipses may be skewed in the estimated "tangent space" direction of the point sample. But none of these are the same as what you propose. – Henry Adams Jan 29 '19 at 16:34