Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter January 25, 2020

A note on norms of signed sums of vectors

  • Giorgos Chasapis EMAIL logo and Nikos Skarmogiannis
From the journal Advances in Geometry


Improving a result of Hajela, we show for every function f with limn→∞f(n) = ∞ and f(n) = o(n) that there exists n0 = n0(f) such that for every nn0 and any S ⊆ {–1, 1}n with cardinality |S| ⩽ 2n/f(n) one can find orthonormal vectors x1, …, xn ∈ ℝn satisfying ε1x1++εnxnclogf(n) for all (ε1, …, εn) ∈ S. We obtain analogous results in the case where x1, …, xn are independent random points uniformly distributed in the Euclidean unit ball B2n or in any symmetric convex body, and the n-norm is replaced by an arbitrary norm on ℝn.

Funding statement: The research of the first named author is supported by the National Scholarship Foundation (IKY), sponsored by the act “Scholarship grants for second-degree graduate studies”, from resources of the operational program “Manpower Development, Education and Life-long Learning”, 2014-2020, co-funded by the European Social Fund (ESF) and the Greek state. The second named author is supported by a PhD Scholarship from the Hellenic Foundation for Research and Innovation(ELIDEK); research number 70/3/14547.

  1. Communicated by: M. Henk


We would like to thank Apostolos Giannopoulos for useful discussions and the anonymous referees for their comments and valuable suggestions that helped to improve the presentation of the results of this article.


[1] S. Artstein-Avidan, A. Giannopoulos, V. D. Milman, Asymptotic geometric analysis. Part I, volume 202 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2015. MR3331351 Zbl 1337.5200110.1090/surv/202Search in Google Scholar

[2] W. Banaszczyk, Balancing vectors and convex bodies. Studia Math. 106 (1993), 93–100. MR1226426 Zbl 0810.4600910.4064/sm-106-1-93-100Search in Google Scholar

[3] W. Banaszczyk, Balancing vectors and Gaussian measures of n-dimensional convex bodies. Random Structures Algorithms12 (1998), 351–360. MR1639752 Zbl 0958.5200410.1002/(SICI)1098-2418(199807)12:4<351::AID-RSA3>3.0.CO;2-SSearch in Google Scholar

[4] N. Bansal, D. Dadush, S. Garg, An algorithm for Komlós conjecture matching Banaszczyk’s bound. In: 57th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2016, 788–799, IEEE Computer Soc., Los Alamitos, CA 2016. MR3631042 Zbl 0705139810.1109/FOCS.2016.89Search in Google Scholar

[5] I. Bárány, On the power of linear dependencies. In: Building bridges, volume 19 of Bolyai Soc. Math. Stud., 31–45, Springer 2008. MR2484636 Zbl 1160.1500110.1007/978-3-540-85221-6_1Search in Google Scholar

[6] A. Barvinok, Measure concentration. Lecture notes, University of Michigan, 2005, available at in Google Scholar

[7] W. Beckner, Inequalities in Fourier analysis. Ann. of Math. (2) 102 (1975), 159–182. MR0385456 Zbl 0338.4201710.2307/1970980Search in Google Scholar

[8] H. J. Brascamp, E. H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions. Advances in Math. 20 (1976), 151–173. MR0412366 Zbl 0339.2602010.1007/978-3-642-55925-9_35Search in Google Scholar

[9] S. Brazitikos, A. Giannopoulos, P. Valettas, B.-H. Vritsiou, Geometry of isotropic convex bodies, volume 196 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2014. MR3185453 Zbl 1304.5200110.1090/surv/196Search in Google Scholar

[10] E. Gluskin, V. Milman, Geometric probability and random cotype 2. In: Geometric aspects of functional analysis, volume 1850 of Lecture Notes in Math., 123–138, Springer 2004. MR2087156 Zbl 1087.4600610.1007/978-3-540-44489-3_12Search in Google Scholar

[11] E. D. Gluskin, Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces. (Russian) Mat. Sb. (N.S.) 136(178) (1988), 85–96. English translation: Math. USSR–Sb. 64 (1989), no. 1, 85–96. MR945901 Zbl 0668.52002Search in Google Scholar

[12] D. Hajela, On a conjecture of Kömlos about signed sums of vectors inside the sphere. European J. Combin. 9 (1988), 33–37. MR938820 Zbl 0646.0501310.1016/S0195-6698(88)80024-6Search in Google Scholar

[13] B. Klartag, R. Vershynin, Small ball probability and Dvoretzky’s theorem. Israel J. Math. 157 (2007), 193–207. MR2342445 Zbl 1120.4600310.1007/s11856-006-0007-1Search in Google Scholar

[14] R. Latała, K. Oleszkiewicz, Small ball probability estimates in terms of widths. Studia Math. 169 (2005), 305–314. MR2140804 Zbl 1073.6004310.4064/sm169-3-6Search in Google Scholar

[15] M. Ledoux, M. Talagrand, Probability in Banach spaces. Springer 1991. MR1102015 Zbl 0748.6000410.1007/978-3-642-20212-4Search in Google Scholar

[16] A. Lytova, K. Tikhomirov, The variance of the pn-norm of the Gaussian vector, and Dvoretzky’s theorem. Algebra i Analiz30 (2018), 107–139 and St. Petersbg. Math. J. 30 (2019), no. 4, 699–722. MR3851373 Zbl 07063280Search in Google Scholar

[17] V. D. Milman, G. Schechtman, An “isomorphic” version of Dvoretzky’s theorem. C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 541–544. MR1356550 Zbl 0836.46007Search in Google Scholar

[18] G. Paouris, P. Valettas, A Gaussian small deviation inequality for convex functions. Ann. Probab. 46 (2018), 1441–1454. MR3785592 Zbl 0689477810.1214/17-AOP1206Search in Google Scholar

[19] G. Paouris, P. Valettas, J. Zinn, Random version of Dvoretzky’s theorem in pn. Stochastic Process. Appl. 127 (2017), 3187–3227. MR3692312 Zbl 1397.4601110.1016/ in Google Scholar

[20] J. Spencer, Six standard deviations suffice. Trans. Amer. Math. Soc. 289 (1985), 679–706. MR784009 Zbl 0577.0501810.1090/S0002-9947-1985-0784009-0Search in Google Scholar

[21] J. Spencer, Balancing vectors in the max norm. Combinatorica6 (1986), 55–65. MR856644 Zbl 0593.9011010.1007/BF02579409Search in Google Scholar

[22] J. Spencer, Ten lectures on the probabilistic method, volume 64 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, PA 1994. MR1249485 Zbl 0822.05060Search in Google Scholar

[23] A. Szankowski, On Dvoretzky’s theorem on almost spherical sections of convex bodies. Israel J. Math. 17 (1974), 325–338. MR0350388 Zbl 0288.5200210.1007/BF02756881Search in Google Scholar

Received: 2018-06-27
Revised: 2019-01-22
Published Online: 2020-01-25
Published in Print: 2021-01-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 27.3.2023 from
Scroll Up Arrow