[1] Adamczak, R., Litvak, A. E., Pajor, A. & Tomczak-Jaegermann, N. Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles, J. Amer. Math. Soc. 23, 2 2010, 535–561.Google Scholar

[2] Adamczak, R., Litvak, A. E., Pajor, A. & Tomczak-Jaegermann, N. Sharp bounds on the rate of convergence of the empirical covariance matrix, C. R. Math. Acad.Sci. Paris 349, 3-4 2011, 195–200.Google Scholar

[3] Alonso-Guttiérez, D. & Bastero, J. The variance conjecture on hyperplane projections of _np balls, https://arxiv.org/abs/1610.04023 (preprint).

[4] Anttila,M., Ball, K. & Perissinaki, I. The central limit problem for convex bodies, Trans. Amer. Math. Soc. 355, 12 2003, 4723–4735.(electronic).Google Scholar

[5] Bandeira, A. S. & van Handel, R. Sharp nonasymptotic bounds on the norm of random matrices with independent entries, Ann. Probab. 44, 4 2016, 2479– 2506.Google Scholar

[6] Berinde, R., Gilbert, A., Indyk, P. H. K. & Strauss, M. Combining geometry and combinatorics: a unified approach to sparse signal recovery, in Communication, Control, and Computing, 2008 46th Annual Allerton Conference on, IEEE 2008, 798–805.Google Scholar

[7] Bourgain, J. Bounded orthogonal systems and the (p)-set problem, Acta Math. 162, 3-4 1989, 227–245.Google Scholar

[8] Bourgain, J. Random points in isotropic convex sets, in Convex Geometric Analysis (Berkeley, CA, 1996), Mathematical Sciences Research Institute Publications 34, Cambridge University Press 1999, 53–58.

[9] Bourgain, J. An improved estimate in the restricted isometry problem, in Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics 2116, Springer 2014, 65–70.

[10] Buser, P. A note on the isoperimetric constant, Ann. Sci. Éc. Norm. Super. (4) 15, 2 1982, 213–230.Google Scholar

[11] CandèsE. J, Romberg, J. K. & Tao, T. Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math. 59, 8 2006, 1207–1223.Google Scholar

[12] Candès, E. J. & Tao, T. Decoding by linear programming, IEEE Trans. Inform.Theory 51, 12 2005, 4203–4215.Google Scholar

[13] Chafaï, D., Guédon, O., Lecué, G. & Pajor, A. Interactions Between Compressed Sensing Random Matrices and High Dimensional Geometry, Panoramas et Synthèses 37, Société Mathématique de France 2012.

[14] Dirksen, S., Lecué, G. & Rauhut, H. On the gap between restricted isometry properties and sparse recovery conditions, https:doi.org/10.1109/TIT.2016.2570244, to appear in IEEE Trans. Inform. Theory.

[15] Donoho, D. L., Compressed sensing, IEEE Trans. Inform. Theory 52, 4 2006, 1289–1306.Google Scholar

[16] Eldan, R. Thin shell implies spectral gap up to polylog via a stochastic localization scheme, Geom. Funct. Anal. 23, 2 2013, 532–569.Google Scholar

[17] Fleury, B. Concentration in a thin Euclidean shell for log-concave measures, J.Funct. Anal. 259, 4 2010, 832–841.Google Scholar

[18] Fleury, B. Poincaré inequality in mean value for Gaussian polytopes, Probab.Theory Related Fields 152, 1-2 2012, 141–178.Google Scholar

[19] Fleury, B., Guédon, O. & Paouris, G. A stability result for mean width of Lp-centroid bodies, Adv. Math. 214, 2 2007, 865–877.Google Scholar

[20] Foucart, S. & Rauhut, H. A Mathematical Introduction to Compressive Sensing, Applied and Numerical Harmonic Analysis. Birkhäuser/Springer 2013.

[21] Friedland, O. & Guédon, O. Sparsity and non-Euclidean embeddings, Israel J.Math. 197, 1 2013, 329–345.Google Scholar

[22] Giannopoulos, A. A. & Milman, V. D., Concentration property on probability spaces, Adv. Math. 156, 1 2000, 77–106.Google Scholar

[23] Gozlan, N., Roberto, C. & Samson, P.-M., From dimension free concentration to the Poincaré inequality, Calc. Var. Partial Differential Equations 52, 3-4 2015, 899–925.Google Scholar

[24] Guédon, O. Kahane–Khinchine type inequalities for negative exponent, Mathematika 46, 1 1999, 165–173.Google Scholar

[25] Guédon, O., Litvak, A. E., Pajor, A. & Tomczak-Jaegermann, N. On the interval of fluctuation of the singular values of random matrices, https://arxiv.org/ abs/1509.02322 (preprint), to appear in J. Eur. Math. Soc. (JEMS).

[26] Guédon, O. & Milman, E. Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures, Geom. Funct. Anal. 21, 5 2011, 1043–1068.Google Scholar

[27] Haviv, I. & Regev, O. The restricted isometry property of subsampled Fourier matrices, https://arxiv.org/abs/1507.01768 (preprint).

[28] Kannan, R., Lovász, L. & Simonovits, M. Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13, 3-4 1995, 541–559.Google Scholar

[29] Kannan, R., Lovász, L. & Simonovits, M. Random walks and an O∗ (n5) volume algorithm for convex bodies, Random Structures Algorithms 11, 1 1997, 1–50.Google Scholar

[30] Klartag, B. On convex perturbations with a bounded isotropic constant, Geom.Funct. Anal. 16, 6 2006, 1274–1290.Google Scholar

[31] Klartag, B. A central limit theorem for convex sets, Invent. Math. 168, 1 2007, 91–131.Google Scholar

[32] Klartag, B. Power-law estimates for the central limit theorem for convex sets, J. Funct. Anal. 245, 1 2007, 284–310.Google Scholar

[33] Kolesnikov, A. V. & Milman, E. The KLS isoperimetric conjecture for generalized Orlicz balls, https://arxiv.org/abs/1610.06336 (preprint).

[34] Latała, R. Some estimates of norms of random matrices, Proc. Amer. Math. Soc. 133, 5 2005, 1273–1282.Google Scholar

[35] Latała, R. & Wojtaszczyk, J. O. On the infimum convolution inequality, Studia Math. 189, 2 2008, 147–187.Google Scholar

[36] Ledoux, M. A simple analytic proof of an inequality by P.Buser, Proc. Amer.Math. Soc. 121, 3 1994, 951–959.Google Scholar

[37] Lee, Y. T. & Vempala, S. S., Eldan's stochastic localization and the KLS hyperplane conjecture: an improved lower bound for expansion, https://arxiv.org/abs/1612.01507 (preprint).

[38] Mendelson, S. & Paouris, G. On the singular values of random matrices, J. Eur.Math. Soc. (JEMS) 16, 4 2014, 823–834.Google Scholar

[39] Milman, E. On the role of convexity in isoperimetry, spectral gap and concentration, Invent. Math. 177, 1 2009, 1–43.Google Scholar

[40] Milman, V. D. & Schechtman, G. Asymptotic Theory of Finite-Dimensional Normed Spaces , Lecture Notes in Mathematics 1200, Springer (1986), With an appendix by M. Gromov.

[41] Oymak, S. & Tropp, J. A., Universality laws for randomized dimension reduction, with applications, https://arxiv.org/abs/1511.09433 (preprint).

[42] Paouris, G. Concentration of mass on convex bodies, Geom. Funct. Anal. 16, 5 2006, 1021–1049.Google Scholar

[43] Rudelson, M. Random vectors in the isotropic position, J. Funct. Anal. 164, 1 1999, 60–72.Google Scholar

[44] Rudelson, M. & Vershynin, R. The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218, 2 2008, 600–633.Google Scholar

[45] Rudelson, M. & Vershynin, R. On sparse reconstruction from Fourier and Gaussian measurements, Comm. Pure Appl. Math. 61, 8 2008, 1025–1045.Google Scholar

[46] Rudelson, M. & Vershynin, R. Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62, 12 2009, 1707–1739.Google Scholar

[47] Seginer, Y. The expected norm of random matrices, Combin. Probab. Comput. 9, 2 2000, 149–166.Google Scholar

[48] Sodin, S. An isoperimetric inequality on the lp balls, Ann. Inst. Henri Poincaré Probab. Stat. 44, 2 2008, 362–373.Google Scholar

[49] Tikhomirov, K. Sample covariance matrices of heavy-tailed distributions, https://arxiv.org/abs/1606.03557 (preprint), to appear in Int. Math. Res. Not.IMRN.

[50] Tikhomirov, K. & Youssef, P. When does a discrete-time random walk in Rn absorb the origin into its convex hull? https://arxiv.org/abs/1410.0458 (preprint), to appear in Ann. Probab.

[51] van Handel, R. On the spectral norm of Gaussian random matrices, https://arxiv.org/abs/1502.05003 (preprint), to appear in Trans. Amer. Math. Soc.

[52] Vempala, S. S., Recent progress and open problems in algorithmic convex geometry, in 30th International Conference on Foundations of Software Technology and Theoretical Computer Science, LIPIcs: Leibniz International Proceedings in Informatics 8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010), pp. 42–64.