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**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**

**Abstract**:

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.