Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


IMPACT FACTOR 2018: 0.789

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.329
Source Normalized Impact per Paper (SNIP) 2018: 0.908

Mathematical Citation Quotient (MCQ) 2018: 0.53

Online
ISSN
1615-7168
See all formats and pricing
More options …
Volume 17, Issue 1

Issues

Mean width of random perturbations of random polytopes

David Alonso-Gutiérrez / Joscha Prochno
Published Online: 2017-02-15 | DOI: https://doi.org/10.1515/advgeom-2016-0032

Abstract

We prove some “high probability” results on the expected value of the mean width for random perturbations of random polytopes. The random perturbations are considered for Gaussian random vectors and uniform distributions on pN-balls and the unit sphere.

Keywords: Random polytope; random perturbation; mean width; Orlicz function

MSC 2010: 52A22; 52A23; 05D40

Communicated by: M. Henk

References

  • [1]

    D. Alonso-Gutiérrez, A. E. Litvak, N. Tomczak-Jaegermann, On the isotropic constant of random polytopes. J. Geom. Anal. 26 (2016), 645–662. MR3441532 Zbl 1338.52008Google Scholar

  • [2]

    D. Alonso-Gutiérrez, J. Prochno, Estimating support functions of random polytopes via Orlicz norms. Discrete Comput. Geom. 49 (2013), 558–588. MR3038530 Zbl 1279.60021Google Scholar

  • [3]

    D. Alonso-Gutiérrez, J. Prochno, On the Gaussian behavior of marginals and the mean width of random polytopes. Proc. Amer. Math. Soc. 143 (2015), 821–832. MR3283668 Zbl 1310.52006Google Scholar

  • [4]

    I. Bárány, A note on Sylvester’s four-point problem. Studia Sci. Math. Hungar. 38 (2001), 73–77. MR1877770 Zbl 1002.60008Google Scholar

  • [5]

    N. Dafnis, A. Giannopoulos, A. Tsolomitis, Asymptotic shape of a random polytope in a convex body. J. Funct. Anal. 257 (2009), 2820–2839. MR2559718 Zbl 1221.52009Google Scholar

  • [6]

    N. Dafnis, A. Giannopoulos, A. Tsolomitis, Quermaßintegrals and asymptotic shape of random polytopes in an isotropic convex body. Michigan Mathematical Journal 62 (2013), 59–79. MR3049297 Zbl 1279.52010Google Scholar

  • [7]

    P. Diaconis, D. Freedman, A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), 397–423. MR898502 Zbl 0619.60039Google Scholar

  • [8]

    Y. Gordon, A. Litvak, C. Schütt, E. Werner, Orlicz norms of sequences of random variables. Ann. Probab. 30 (2002), 1833–1853. MR1944007 Zbl 1016.60008Google Scholar

  • [9]

    Y. Gordon, A. Litvak, C. Schütt, E. Werner, Minima of sequences of Gaussian random variables. C. R. Math. Acad. Sci. Paris 340 (2005), 445–448. MR2135327 Zbl 1064.60030Google Scholar

  • [10]

    Y. Gordon, A. E. Litvak, C. Schütt, E. Werner, Uniform estimates for order statistics and Orlicz functions. Positivity 16 (2012), 1–28. MR2892571 Zbl 1263.62090Google Scholar

  • [11]

    M. Gromov, V. D. Milman, Generalization of the spherical is operimetric inequality to uniformly convex Banach spaces. Compositio Math. 62 (1987), 263–282. MR901393 Zbl 0623.46007Google Scholar

  • [12]

    B. Klartag, Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245 (2007), 284–310. MR2311626 Zbl 1140.52004Google Scholar

  • [13]

    M. A. Krasnoselski, Y. B. Rutickii, Convex Functions and Orlicz Spaces. P. Noordhof LTD., Groningen (1961).Google Scholar

  • [14]

    M. Ledoux, The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2001. MR1849347 Zbl 0995.60002Google Scholar

  • [15]

    A. E. Litvak, A. Pajor, M. Rudelson, N. Tomczak-Jaegermann, Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195 (2005), 491–523. MR2146352 Zbl 1077.15021Google Scholar

  • [16]

    S. Mendelson, A. Pajor, On singular values of matrices with independent rows. Bernoulli 12 (2006), 761–773. MR2265341 Zbl 1138.60328Google Scholar

  • [17]

    V. D. Milman, G. Schechtman, Asymptotic theory of finite-dimensional normed spaces. Springer 1986. MR856576 Zbl 0606.46013Google Scholar

  • [18]

    G. Paouris, Concentration of mass on isotropic convex bodies. C. R. Math. Acad. Sci. Paris 342 (2006), 179–182. MR2198189 Zbl 1087.52002Google Scholar

  • [19]

    G. Paouris, E. M. Werner, Relative entropy of cone measures and Lp centroid bodies. Proc. Lond. Math. Soc. (3) 104 (2012), 253–286. MR2880241 Zbl 1246.52008Google Scholar

  • [20]

    J. Prochno, S. Riemer, On the maximum of random variables on product spaces. Houston J. Math. 39 (2013), 1301–1311. MR3164717 Zbl 1290.46011Google Scholar

  • [21]

    M. M. Rao, Z. D. Ren, Theory of Orlicz spaces. Dekker 1991. MR1113700 Zbl 0724.46032Google Scholar

  • [22]

    A. Rényi, R. Sulanke, Über die konvexe Hülle von n zufällig gewählten Punkten. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1963), 75–84 (1963). MR0156262 Zbl 0118.13701Google Scholar

  • [23]

    A. Rényi, R. Sulanke, Über die konvexe Hülle von n zufällig gewählten Punkten. II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 138–147 (1964). MR0169139 Zbl 0126.34103Google Scholar

  • [24]

    A. Rényi, R. Sulanke, Zufällige konvexe Polygone in einem Ringgebiet. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 (1968), 146–157. MR0229272 Zbl 0162.49502Google Scholar

  • [25]

    G. Schechtman, J. Zinn, On the volume of the intersection of two Lpn balls. Proc. Amer. Math. Soc. 110 (1990), 217–224. MR1015684 Zbl 0704.60017Google Scholar

  • [26]

    G. Schechtman, J. Zinn, Concentration on the lpn ball. In: Geometric aspects of functional analysis, volume 1745 of Lecture Notes in Math., 245–256, Springer 2000. MR1796723 Zbl 0971.46009Google Scholar

  • [27]

    D. A. Spielman, S.-H. Teng, Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM 51 (2004), 385–463. MR2145860 Zbl 1192.90120Google Scholar

  • [28]

    J. J. Sylvester, Question 1491. Educational Times, London, April 1864.Google Scholar

About the article

Received: 2015-02-13

Revised: 2015-03-24

Published Online: 2017-02-15

Published in Print: 2017-01-01


Citation Information: Advances in Geometry, Volume 17, Issue 1, Pages 75–90, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2016-0032.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in