We generalize the concept of the Voronoi path of a line to more general shapes and compute the distortion constant, which describes how it changes volume on the average.  Although initially asked for a Poisson point process, the distortion is a characteristic property of the space rather than the point process.  In other words, the constant ratio of the perimeter of a circle and its pixelation---and the analogous ratios for spheres in three and higher dimensions---hold for all smoothly embedded shapes on average.   This is joint work with Anton Nikitenko.