Sylvester's four point problem for ball-convex bodies

Source abstract

We investigate Sylvester's classical four-point problem for ball-convex bodies, where convex hulls are replaced by intersections of balls of a fixed radius $R$. We show that the Efron--Buchta identities persist in this framework, so that the distribution of the number of vertices of the $R$-ball-convex hull of $n$ uniform random points is determined by the moments of the volumes of the hulls of its subsamples, and vice versa. 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. In particular, this yields an isoperimetric inequality for the ball-convex analogue of the affine length of the boundary curve. For the unit disc, we can express the Sylvester probabilities using dilogarithmic functions of the radius $R$, and give explicit values at $R=1$. As $R\to\infty$, these recover the classical Sylvester probabilities. As another consequence, we determine the exact distribution function of the circumradius of three uniform random points in he disc on $[1,\infty)$.