Random approximation of convex bodies in Hausdorff metric (2404.02870v1)
Abstract: While there is extensive literature on approximation, deterministic as well as random, of general convex bodies $K$ in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding random results in the Hausdorff distance when the approximant $K_n$ is given by the convex hull of $n$ independent random points chosen uniformly on the boundary or in the interior of $K$. When $K$ is a polygon and the points are chosen on its boundary, we determine the exact limiting behavior of the expected Hausdorff distance between a polygon as $n\to\infty$. From this we derive the behavior of the asymptotic constant for a regular polygon in the number of vertices.
- I. Bárány. Intrinsic volumes and f𝑓fitalic_f-vectors of random polytopes. Math. Ann., 285(4):671–699, 1989.
- I. Bárány. Random polytopes in smooth convex bodies. Mathematika, 39(1):81–92, 1992.
- I. Bárány and C. Buchta. Random polytopes in a convex polytope, independence of shape, and concentration of vertices. Math. Ann., 297(3):467–497, 1993.
- On the Hausdorff distance between a convex set and an interior random convex hull. Adv. in Appl. Probab., 30(2):295–316, 1998.
- V.-E. Brunel. Uniform behaviors of random polytopes under the Hausdorff metric. Bernoulli, 25(3):1770–1793, 2019.
- L. Dümbgen and G. Walther. Rates of convergence for random approximations of convex sets. Adv. in Appl. Probab., 28(2):384–393, 1996.
- S. Glasauer and R. Schneider. Asymptotic approximation of smooth convex bodies by polytopes. Forum Math., 8(3):363–377, 1996.
- Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, seventh edition, 2007.
- G. Last and M. Penrose. Lectures on the Poisson process, volume 7 of Institute of Mathematical Statistics Textbooks. Cambridge University Press, Cambridge, 2018.
- Best and random approximation of a convex body by a polytope. J. Complexity, 71:Paper No. 101652, 19, 2022.
- M. Reitzner. Random points on the boundary of smooth convex bodies. Trans. Amer. Math. Soc., 354(6):2243–2278, 2002.
- The convex hull of random points on the boundary of a simple polytope. Discrete Comput. Geom., 69(2):453–504, 2023.
- A. Rényi and R. Sulanke. Über die konvexe Hülle von n𝑛nitalic_n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 2:75–84 (1963), 1963.
- A. Rényi and R. Sulanke. Über die konvexe Hülle von n𝑛nitalic_n zufällig gewählten Punkten. II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3:138–147 (1964), 1964.
- R. Schneider. Random approximation of convex sets*. Journal of Microscopy, 151(3):211–227, 1988.
- C. Schütt. Random polytopes and affine surface area. Math. Nachr., 170:227–249, 1994.
- C. Schütt and E. M. Werner. Polytopes with vertices chosen randomly from the boundary of a convex body. In Geometric aspects of functional analysis, volume 1807 of Lecture Notes in Math., pages 241–422. Springer, Berlin, 2003.
- G. Walther. On a generalization of Blaschke’s rolling theorem and the smoothing of surfaces. Math. Methods Appl. Sci., 22(4):301–316, 1999.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.