标题： 在具有更高平滑度的平面凸体中的广义圆盘多边形 显示英文标题

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

摘要： 我们证明了在边界足够光滑的平面凸体中，随机$L$-凸多边形的顶点数和遗漏面积的期望值的幂级数展开式。随机$L$-凸多边形是包含独立同分布均匀随机点的固定凸集$L$的所有平移的交集，这些随机点来自合适的平面凸体$K$。我们的结果扩展了 Fodor、Papvári 和 Vígh（2020）以及 Fodor 和 Montenegro（2024）证明的渐近公式，并对 Schütt、Werner 和 Yalikun（2025）研究的$L$-凸浮动物体和相对仿射表面面积产生了影响。 显示英文摘要

摘要： 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).