On Generalized Disc-Polygons in Plane Convex Bodies With a Higher Degree of Smoothness

Abstract

Abstract We prove power series expansions for the expectations of the number of vertices and missed area of random L -convex polygons in planar convex bodies with sufficiently smooth boundaries. Random L -convex polygons arise as the intersection of all translates of a fixed convex set L that contain i.i.d. uniform random points from a suitable plane convex body K . Our results extend the asymptotic formulas proved in Fodor, Papvári and Vígh [11] and Fodor and Montenegro [9], and have consequences about L -convex floating bodies and relative affine surface area that were investigated by Schütt, Werner and Yalikun in [19].