A simplified proof of CLT for convex bodies

Fresen, Daniel

doi:10.1090/proc/15000

A simplified proof of CLT for convex bodies

Author: Daniel J. Fresen
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
Published electronically: October 1, 2020
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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.

[1] 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, https://doi.org/10.1090/S0002-9947-03-03085-X
[2] S. G. Bobkov, On concentration of distributions of random weighted sums, Ann. Probab. 31 (2003), no. 1, 195–215. MR 1959791, https://doi.org/10.1214/aop/1046294309
[3] Ulrich Brehm and Jürgen Voigt, Asymptotics of cross sections for convex bodies, Beiträge Algebra Geom. 41 (2000), no. 2, 437–454. MR 1801435
[4] Persi Diaconis and David Freedman, Asymptotics of graphical projection pursuit, Ann. Statist. 12 (1984), no. 3, 793–815. MR 751274, https://doi.org/10.1214/aos/1176346703
[5] 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, https://doi.org/10.1016/j.aim.2007.03.008
[6] B. Fleury, Concentration in a thin Euclidean shell for log-concave measures, J. Funct. Anal. 259 (2010), no. 4, 832–841. MR 2652173, https://doi.org/10.1016/j.jfa.2010.04.019
[7] Daniel J. Fresen, Explicit Euclidean embeddings in permutation invariant normed spaces, Adv. Math. 266 (2014), 1–16. MR 3262353, https://doi.org/10.1016/j.aim.2014.07.017
[8] 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, https://doi.org/10.1007/s00039-011-0136-5
[9] B. Klartag, A central limit theorem for convex sets, Invent. Math. 168 (2007), no. 1, 91–131. MR 2285748, https://doi.org/10.1007/s00222-006-0028-8
[10] B. Klartag, Power-law estimates for the central limit theorem for convex sets, J. Funct. Anal. 245 (2007), no. 1, 284–310. MR 2311626, https://doi.org/10.1016/j.jfa.2006.12.005
[11] Bo’az Klartag, High-dimensional distributions with convexity properties, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2010, pp. 401–417. MR 2648334, https://doi.org/10.4171/077-1/17
[12] 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, https://doi.org/10.1109/FOCS.2017.96
[13] Y. T. Lee and S. Vempala, The Kannan-Lovász-Simonovits conjecture, arXiv:1807.03465.
[14] László Lovász and Santosh Vempala, The geometry of logconcave functions and sampling algorithms, Random Structures Algorithms 30 (2007), no. 3, 307–358. MR 2309621, https://doi.org/10.1002/rsa.20135
[15] 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
[16] Gideon Schechtman, Euclidean sections of convex bodies, Asymptotic geometric analysis, Fields Inst. Commun., vol. 68, Springer, New York, 2013, pp. 271–288. MR 3076155, https://doi.org/10.1007/978-1-4614-6406-8_12