Abstract
Given an arbitrary 1-Lipschitz function f on the torus \(\mathbb{T}^{n}\), we find a k-dimensional subtorus \(M \subseteq \mathbb{T}^{n}\), parallel to the axes, such that the restriction of f to the subtorus M is nearly a constant function. The k-dimensional subtorus M is selected randomly and uniformly. We show that when \(k \leq c\log n/(\log \log n +\log 1/\varepsilon )\), the maximum and the minimum of f on this random subtorus M differ by at most \(\varepsilon\), with high probability.
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Acknowledgements
We would like to thank Vladimir Pestov for his interest in this work. The first-named author was partially supported by ISF grants 701/08 and 1447/12. The second-named author was supported by a grant from the European Research Council (ERC). The third-named author was supported by ISF grant 387/09 and by BSF grant 2006079.
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Faifman, D., Klartag, B., Milman, V. (2014). On the Oscillation Rigidity of a Lipschitz Function on a High-Dimensional Flat Torus. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_10
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DOI: https://doi.org/10.1007/978-3-319-09477-9_10
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