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This is a Quantitative Study study.

October 13, 2025Open Access

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

Key Points

The study reveals power series expansions for the expected number of vertices in random L-convex polygons.
It extends prior asymptotic formulas, enhancing understanding of convex structures and their properties.
Analysis incorporates random L-convex polygons formed by random points in planar convex bodies, with implications for floating bodies.
Results provide insights into the relationship between smooth boundaries and geometric probabilities in convex forms.

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 (2020) and Fodor and Montenegro (2024), and have consequences about L-convex floating bodies and relative affine surface area that were investigated by Schütt, Werner and Yalikun (2025).

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