On generalized disc-polygons in plane convex bodies with a higher degree of smoothness

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, Papv\'ari and V\'igh (2020) and Fodor and Montenegro (2024), and have consequences about $L$-convex floating bodies and relative affine surface area that were investigated by Sch\"utt, Werner and Yalikun (2025).