Random Spherical Disc-Polygons in a Spherical Spindle Convex Disc

In this paper, we consider the asymptotic behaviour of the expectation of the number of vertices of a uniform random spherical disc-polygon. This provides a connection between the corresponding results in spherical convexity, and in Euclidean spindle-convexity, where the expectation tends to the same constant. We also extend the result to a more general case, where the random points generating the uniform random disc-polygon are chosen from spherical convex disc with smooth boundary.