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A note on norms of signed sums of vectors

Giorgos Chasapis and Nikos Skarmogiannis
From the journal Advances in Geometry

Abstract

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

Acknowledgements

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.

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

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