## A simplified proof of CLT for convex bodies

HTML articles powered by AMS MathViewer

- by Daniel J. Fresen PDF
- Proc. Amer. Math. Soc.
**149**(2021), 345-354 Request permission

## Abstract:

We present a short proof of Klartag’s central limit theorem for convex bodies, using only the most classical facts about log-concave functions. An appendix is included where we give the proof that the thin shell implies CLT. The paper is accessible to anyone.## References

- Milla Anttila, Keith Ball, and Irini Perissinaki,
*The central limit problem for convex bodies*, Trans. Amer. Math. Soc.**355**(2003), no. 12, 4723–4735. MR**1997580**, DOI 10.1090/S0002-9947-03-03085-X - S. G. Bobkov,
*On concentration of distributions of random weighted sums*, Ann. Probab.**31**(2003), no. 1, 195–215. MR**1959791**, DOI 10.1214/aop/1046294309 - Ulrich Brehm and Jürgen Voigt,
*Asymptotics of cross sections for convex bodies*, Beiträge Algebra Geom.**41**(2000), no. 2, 437–454. MR**1801435** - Persi Diaconis and David Freedman,
*Asymptotics of graphical projection pursuit*, Ann. Statist.**12**(1984), no. 3, 793–815. MR**751274**, DOI 10.1214/aos/1176346703 - B. Fleury, O. Guédon, and G. Paouris,
*A stability result for mean width of $L_p$-centroid bodies*, Adv. Math.**214**(2007), no. 2, 865–877. MR**2349721**, DOI 10.1016/j.aim.2007.03.008 - B. Fleury,
*Concentration in a thin Euclidean shell for log-concave measures*, J. Funct. Anal.**259**(2010), no. 4, 832–841. MR**2652173**, DOI 10.1016/j.jfa.2010.04.019 - Daniel J. Fresen,
*Explicit Euclidean embeddings in permutation invariant normed spaces*, Adv. Math.**266**(2014), 1–16. MR**3262353**, DOI 10.1016/j.aim.2014.07.017 - Olivier Guédon and Emanuel Milman,
*Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures*, Geom. Funct. Anal.**21**(2011), no. 5, 1043–1068. MR**2846382**, DOI 10.1007/s00039-011-0136-5 - B. Klartag,
*A central limit theorem for convex sets*, Invent. Math.**168**(2007), no. 1, 91–131. MR**2285748**, DOI 10.1007/s00222-006-0028-8 - B. Klartag,
*Power-law estimates for the central limit theorem for convex sets*, J. Funct. Anal.**245**(2007), no. 1, 284–310. MR**2311626**, DOI 10.1016/j.jfa.2006.12.005 - Bo’az Klartag,
*High-dimensional distributions with convexity properties*, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2010, pp. 401–417. MR**2648334**, DOI 10.4171/077-1/17 - Yin Tat Lee and Santosh S. Vempala,
*Eldan’s stochastic localization and the KLS hyperplane conjecture: an improved lower bound for expansion*, 58th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2017, IEEE Computer Soc., Los Alamitos, CA, 2017, pp. 998–1007. MR**3734299**, DOI 10.1109/FOCS.2017.96 - Y. T. Lee and S. Vempala,
*The Kannan-Lovász-Simonovits conjecture*, arXiv:1807.03465. - László Lovász and Santosh Vempala,
*The geometry of logconcave functions and sampling algorithms*, Random Structures Algorithms**30**(2007), no. 3, 307–358. MR**2309621**, DOI 10.1002/rsa.20135 - V. D. Milman,
*A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies*, Funkcional. Anal. i Priložen.**5**(1971), no. 4, 28–37 (Russian). MR**0293374** - Gideon Schechtman,
*Euclidean sections of convex bodies*, Asymptotic geometric analysis, Fields Inst. Commun., vol. 68, Springer, New York, 2013, pp. 271–288. MR**3076155**, DOI 10.1007/978-1-4614-6406-8_{1}2

## Additional Information

**Daniel J. Fresen**- Affiliation: Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield, Pretoria 0028, South Africa
- MR Author ID: 977451
- Email: daniel.fresen@up.ac.za
- Received by editor(s): July 19, 2019
- Received by editor(s) in revised form: December 7, 2019
- Published electronically: October 1, 2020
- Additional Notes: Funding received from the University of Pretoria Research Development Programme
- Communicated by: Zhen-Qing Chen
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 345-354 - MSC (2010): Primary 52A20, 52A23, 60F05; Secondary 26B25, 52A38, 62E20
- DOI: https://doi.org/10.1090/proc/15000
- MathSciNet review: 4172610