Average and Expected Distortion of Voronoi Paths and Scapes

The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about \(\tfrac{4}{\pi }\) $\frac{4}{\pi }$ . We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes.