If $\mathscr P\subset \mathbb R^d$ is a discrete point configuration, take the Voronoi diagram of $\mathscr P$ and call $\mathscr P'$ the vertices of that diagram.
I would like to know if configurations such that $\mathscr P = \mathscr P''$ have an established name, and if people studied them.
Some first facts and questions that may make the above property interesting:
- If $\mathscr P$ has the property, then $\mathscr P'$ has it.
- If $\mathscr P$ has the property, then if we join the vertices of $\mathscr P$ to their nearest neighbors in $\mathscr P'$, we form a bipartite graph with edges of equal length. Not all such graphs give $\mathscr P$ satisfying the property, but does there exist a simple extra convexity condition that can be added to get it?
- It seems that $\mathscr P$ that have the property are unbounded, is that true? I could not prove or find a counterexample so far.
- Do there exist $\mathscr P$ that have the property and that have accumulation points?
- Examples: (1) All simple lattices I tried as $\mathscr P$ have the property. Are there lattices that do not have the property? (2) Vertices of rhombic tilings of $\mathbb R^2$ have the property. In particular this includes Penrose tilings.