Higher ($>1$) order Voronoi diagrams in higher ($ > 2$) dimensional spaces

Question

The $k^{th}$ order Voronoi partitioning given a set of $K$ points $X_K$ in $\mathbb{R}^d$ partitions the space into regions such that points in the same region have the same $k$ nearest neighbors in $X_K$.

Would appreciate if anyone could point me in the direction of any resources that discuss the properties of $k^{th}$-order Voronoi diagrams in higher dimensional Euclidean (not just $\mathbb{R}^2$) spaces.

Specifically for the $k^{th}$ order Voronoi diagram in $\mathbb{R}^2$, this paper (Lemma 5) addresses that if two Voronoi cells share an edge, then only the $k^{th}$ nearest neighbor changes. I am interested in knowing if a similar result holds in higher dimensional Euclidean spaces.

Answer 1

Maybe start here:

Agarwal, Pankaj K., Mark De Berg, Jirí Matousek, and Otfried Schwarzkopf. "Constructing levels in arrangements and higher order Voronoi diagrams." SIAM Journal on Computing 27, no. 3 (1998): 654-667. (PDF download.)

See also this book for a more recent survey with many citations:

Aurenhammer, Franz, Rolf Klein, and Der-Tsai Lee. Voronoi diagrams and Delaunay triangulations. World Scientific Publishing Company, 2013. (Publisher link.)