Covering a Sphere with Four Random Circular Caps

TL;DR

This paper investigates the probability that four random spherical caps of large angular radius cover a unit sphere, providing bounds for this probability and exploring a related inscribed tetrahedra problem.

Contribution

It offers new lower bounds for the coverage probability of four large caps on a sphere and examines a dual problem involving inscribed tetrahedra.

Findings

Derived nontrivial lower bounds for coverage probability when cap radius exceeds 84 degrees.

Established that no improvements are currently feasible for the probability bound below 84 degrees.

Explored a dual problem related to inscribed well-centered tetrahedra in the sphere.

Abstract

Let p(w) denote the probability that four random circular caps of angular radius 70deg<w<90deg cover the unit sphere S^2. An exact expression for p(w) is unknown. We give nontrivial lower bounds for p(w) when w>84deg; no improvement on the inequality p(w)>=0 for w<84deg is yet feasible. A dual problem involving randomly inscribed well-centered tetrahedra is also examined.