Ferenc Fodor: Sylvester's four-point problem for ball-convex bodies

We investigate Sylvester's classical problem for ball-convex bodies, where convex hulls are replaced by intersections of balls of a fixed radius R. For three and four uniform random points in an arbitrary planar ball-convex body, we determine the Sylvester probabilities of the R-ball-convex hull. We also give an extension of Groemer's inequality for the expected area of the ball-convex hull of n random points and show that it is minimised by the disc of the same area, which leads to an isoperimetric inequality for the ball-convex analogue of the affine length of the boundary curve.

This is joint work with A. Bakó-Szabó (University of Szeged, Hungary) and Florian Besau (TU Wien, Austria).

The talk will also be broadcast on Zoom.