# Scale dependence of Voronoi path length

Consider a square of fixed size in Euclidean space. Assume the square has been decomposed into Voronoi blocks of a certain average area.

Now assume you can only move along the edges of the Voronoi diagram, and you want to cover a certain distance in the embedding space. Is there a result on how your minimum path length along teh Vornoi edges depends of the fineness of the Voronoi decomposition (i.e. on the average size of the blocks)?

Update: To make this more specific, consider the following example: I have three Voronoi decompositions of average edeg length 0.3, 0.1 and 0.02, and I'm interested in the lengths of the red paths, i.e. the expected minimum path length. In the example, the paths are about the same length, but I don't know whether there is a more general statement known. I have a feeling the the larger cells have fewer longer detours, while the smaller cells have more shorter ones, and the effects cancels out, to some extent.

I assume a "reasonably uniform" point distribution, and the Voronoi cells are much smaller that the surrounding square, so the sides of the square should not have a major influence.

• Can you expand on the question? What do you mean by "minimum path length"? Perhaps the shortest segment on the edge associated with the same Voronoi cell? – mlc Mar 26 '17 at 17:25
• By minimum path length I mean the shortest path along sevetal cells. Say the tesselated square is the unit square, the I'm interested in the path from e.g. the left edge ($x=0$) to the right one ($x=1$). I have a feeling it does not depend on the Voronoi size, but actually I don't know. – Toffomat Mar 27 '17 at 13:50
• If you are inside a cell, how do you get to the edge? Perhaps you should offer more details and make your question specific. – mlc Mar 27 '17 at 14:08