Series expansions for random disc-polygons in smooth plane convex bodies (Q6639537)

This paper deals with asymptotic expansions for expected geometric functionals of certain random approximations of planar convex bodies. Let \(K\) be a convex body in \({\mathbb R}^2\) with a boundary of class \(C^{k+1}\), for some \(k\ge 2\), and positive curvature \(\kappa\). For \(R>\max_{x\in\partial K} 1/\kappa(x)\) and \(n\) independent uniform random points in \(K\), let \(K_n^R\) be the \(R\)-spindle convex hull of these points (the intersection of all circular discs of radius \(R\) containing the points). The main result of the authors says that the expectation of the vertex number satisfies \N\[\N{\mathbb E}(f_0(K_n^R)) = z_1(K)n^{1/3} +0\cdot n^0+\dots+ z_{k-1}n^{-(k-3)/3} +O(n^{-(k-2)/3})\N\]\Nas \(n\to\infty\). The coefficients can be computed explicitly, for example, \N\[\Nz_1(K) = \sqrt{\frac{2}{3A(K)}}\Gamma\left(\frac{5}{3}\right)\int_{\partial K}\left(\kappa(x)-\frac{1}{R}\right)^{1/3}\mathrm{d}x,\N\]\Nwhere \(A\) denotes the area. The higher-order coefficients depend on \(A(K)\), \(R\) and derivatives of the curvature function. A similar result for the missed area can be deduced from this one via an Efron-type identity. The case \(k=1\) was treated earlier, by \textit{F. Fodor} et al. [Adv. Appl. Probab. 46, No. 4, 899--918 (2014; Zbl 1314.52004)]. It is also discussed what happens to the approximation when the radius \(R\) turns to the limits of its range. Moreover, the case when \(K\) is the circular disc of radius \(R\) (which is not covered by the above) is considered in some detail.