The mean width of the oloid and integral geometric applications of it

TL;DR

This paper computes the mean width of the oloid using integral geometry, derives related geometric measures, and explores probabilistic expectations of intersections with moving spheres and ovoids.

Contribution

It provides the first explicit calculation of the oloid's mean width via two methods and applies these results to integral geometric problems involving intersections.

Findings

Mean width of the oloid calculated via mean curvature integral and direct methods

Derived formulas for surface area and volume of the oloid's parallel body

Expected measures of intersections with moving balls and ovoids

Abstract

The oloid is the convex hull of two circles with equal radius in perpendicular planes so that the center of each circle lies on the other circle. We calculate the mean width of the oloid in two ways, first via the integral of mean curvature, and then directly. Using this result, the surface area and the volume of the parallel body are obtained. Furthermore, we derive the expectations of the mean width, the surface area and the volume of the intersections of a fixed oloid and a moving ball, as well as of a fixed and a moving oloid.