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An intrinsic volume metric for the class of convex bodies in $ℝ^{n}$

Florian Besau

Institute of Discrete Mathematics and Geometry, Technische Universität Wien, Vienna, Austria

E-mail Address: [email protected]

Corresponding author.

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and

Steven Hoehner

Department of Mathematics & Computer Science, Longwood University, Farmville, VA 23909, USA

E-mail Address: [email protected]

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https://doi.org/10.1142/S0219199723500062Cited by:1 (Source: Crossref)

Abstract

A new intrinsic volume metric is introduced for the class of convex bodies in $ℝ^{n}$ . As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width approximation of the ball.

Keywords:

AMSC: 52A20, 52A22, 52A27, 52A39, 52B11

References

1. F. Affentranger, The convex hull of random points with spherically symmetric distributions, Rend. Sem. Mat. Univ. Politec. Torino 49 (1991) 359–383. Google Scholar
2. I. Bárány, F. Fodor and V. Vígh, Intrinsic volumes of inscribed random polytopes in smooth convex bodies, Adv. in Appl. Probab. 42 (2010) 605–619. Crossref, Web of Science, Google Scholar
3. F. Besau, S. Hoehner and G. Kur, Intrinsic and dual volume deviations of convex bodies and polytopes, Int. Math. Res. Not. 22 (2021) 17456–17513. Crossref, Google Scholar
4. K. J. Böröczky, F. Fodor and D. Hug, Intrinsic volumes of random polytopes with vertices on the boundary of a convex body, Trans. Amer. Math. Soc. 365 (2013) 785–809. Crossref, Web of Science, Google Scholar
5. K. J. Böröczky, L. M. Hoffmann and D. Hug, Expectation of intrinsic volumes of random polytopes, Period. Math. Hungar. 57 (2008) 143–164. Crossref, Web of Science, Google Scholar
6. S.-S. Chern, On the kinematic formula in integral geometry, J. Math. Mech. 16 (1966) 101–118. Google Scholar
7. A. Florian, On a metric for the class of compact convex sets, Geom. Dedicata 30 (1989) 69–80. Crossref, Web of Science, Google Scholar
8. R. J. Gardner, Geometric Tomography, Encyclopedia of Mathematics and its Applications, Vol. 58 (Cambridge University Press, 2006). Crossref, Google Scholar
9. S. Glasauer and P. M. Gruber, Asymptotic estimates for best and stepwise approximation of convex bodies III, Forum Math. 9 (1997) 383–404. Crossref, Web of Science, Google Scholar
10. H. Groemer, On the symmetric difference metric for convex bodies, Beiträge Algebra Geom. 41 (2000) 107–114. Google Scholar
11. J. Grote, C. Thäle and E. Werner, Surface area deviation between smooth convex bodies and polytopes, Adv. in Appl. Math. 129 (2021) 102218. Crossref, Web of Science, Google Scholar
12. J. Grote and E. Werner, Approximation of smooth convex bodies by random polytopes, Electron. J. Probab. 23 (2018) 1–21. Crossref, Web of Science, Google Scholar
13. P. M. Gruber, Asymptotic estimates for best and stepwise approximation of convex bodies II, Forum Math. 5 (1993) 521–538. Google Scholar
14. H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer, Berlin, 1957). Crossref, Google Scholar
15. S. Hoehner and G. Kur, A concentration inequality for random polytopes, Dirichlet–Voronoi tiling numbers and the geometric balls and bins problem, Discrete Comput. Geom. 65 (2021) 730–763. Crossref, Web of Science, Google Scholar
16. S. Hoehner, B. Li, M. Roysdon and C. Thäle, Asymptotic expected $T$ -functionals of random polytopes with applications to $L_{p}$ surface areas, preprint (2022), arXiv:2202.01353v2. Google Scholar
17. S. Hoehner, C. Schütt and E. Werner, The surface area deviation of the Euclidean ball and a polytope, J. Theoret. Probab. 31 (2018) 244–267. Crossref, Web of Science, Google Scholar
18. Z. Kabluchko, D. Temesvari and C. Thäle, Expected intrinsic volumes and facet numbers of random beta-polytopes, Math. Nachr. 292 (2019) 79–105. Crossref, Web of Science, Google Scholar
19. D. A. Klain and G.-C. Rota, Introduction to Geometric Probability (Cambridge University Press, 1997). Google Scholar
20. G. Kur, Approximation of the Euclidean ball by polytopes with a restricted number of facets, Studia Math. 251 (2020) 111–133. Crossref, Web of Science, Google Scholar
21. M. Ludwig, Asymptotic approximation of smooth convex bodies by general polytopes, Mathematika 46 (1999) 103–125. Crossref, Web of Science, Google Scholar
22. M. Ludwig, C. Schütt and E. M. Werner, Approximation of the Euclidean ball by polytopes, Studia Math. 173 (2006) 1–18. Crossref, Web of Science, Google Scholar
23. E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975) 531–538. Crossref, Web of Science, Google Scholar
24. A. L. G. Mandolesi, Blade products and angles between subspaces, Adv. Appl. Clifford Algebras 31 (2021) 69. Crossref, Web of Science, Google Scholar
25. R. E. Miles, Isotropic random simplices, Adv. in Appl. Probab. 3 (1971) 353–382. Crossref, Google Scholar
26. M. Reitzner, Random points on the boundary of smooth convex bodies, Trans. Amer. Math. Soc. 354 (2002) 2243–2278. Crossref, Web of Science, Google Scholar
27. L. A. Santaló, Integral Geometry and Geometric Probability (Cambridge University Press, Cambridge, 2004). Crossref, Google Scholar
28. R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia of Mathematics and its Applications, 2nd edn. (Cambridge University Press, 2013). Crossref, Google Scholar
29. R. Schneider and W. Weil, Stochastic and Integral Geometry (Springer, Berlin, 2008). Crossref, Google Scholar
30. C. Schütt and E. Werner, Polytopes with vertices chosen randomly from the boundary of a convex body, in Geometric Aspects of Functional Analysis: Israel Seminar 2001–2002, Lecture Notes in Mathematics, Vol. 1807 (Springer, Berlin, Heidelberg, 2003), pp. 241–422. Crossref, Google Scholar
31. G. C. Shephard and R. J. Webster, Metrics for sets of convex bodies, Mathematika 12 (1965) 73–88. Crossref, Web of Science, Google Scholar
32. R. A. Vitale, $L_{p}$ metrics for compact, convex sets, J. Approx. Theory 45 (1985) 280–287. Crossref, Web of Science, Google Scholar
33. J. G. Wendel, Note on the gamma function, Amer. Math. Monthly 55 (1948) 563–564. Crossref, Google Scholar