This was part of Randomness in Topology and its Applications

Distortion, on the Average and in Expectation

Herbert Edelbrunner, ISTA - Institute of Science and Technology Austria
Monday, March 20, 2023

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.