Average and Expected Distortion of Voronoi Paths and Scapes

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