Abstract
Let \(\varepsilon \in (0,1/2)\). We prove that if for some n > 1 and k > 1, a majority of k-dimensional sections of the ball in \(l_{\infty }^{n}\) is \((1+\varepsilon )\)-spherical then necessarily \(k \leq C\varepsilon \ln n/\ln \frac{1} {\varepsilon }\), where C is a universal constant. The bound for k is optimal up to the choice of C.
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References
H.A. David, Order Statistics, 2nd edn. (Wiley, New York, 1981)
A. Dvoretzky, Some results on convex bodies and Banach spaces, in Proceedings of the International Symposium on Linear Spaces, Jerusalem (1961), pp. 123–160
W. Feller, An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. (Wiley, New York, 1968)
Y. Gordon, Some inequalities for Gaussian processes and applications. Israel J. Math. 50(4), 265–289 (1985)
Y. Gordon, A.E. Litvak, A. Pajor, N. Tomczak-Jaegermann, Random \(\varepsilon\)-nets and embeddings in \(l_{\infty }^{N}\). Studia Math. 178(1), 91–98 (2007)
V.D. Milman, A new proof of the theorem of A. Dvoretzky on sections of convex bodies. Funct. Anal. Appl. 5(4), 288–295 (1971)
V.D. Milman, G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Mathematics, vol. 1200 (Springer, Berlin, 1986)
G. Pisier, The Volumes of Convex Bodies and Banach Space Geometry (Cambridge University Press, Cambridge, 1989)
G. Schechtman, Two observations regarding embedding subsets of Euclidean spaces in normed spaces. Adv. Math. 200, 125–135 (2006)
G. Schechtman, The random version of Dvoretzky’s theorem in \(l_{\infty }^{n}\), in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1910 (Springer, Berlin, 2007), pp. 265–270
G. Schechtman, Euclidean sections of convex bodies, in Asymptotic Geometric Analysis, ed. by M. Ludwig et al. Fields Institute Communications, vol. 68 (Springer, New York, 2013), 271–288
K.E. Tikhomirov, Almost Euclidean sections in symmetric spaces and concentration of order statistics. J. Funct. Anal. 265, 2074–2088 (2013)
Acknowledgements
I would like to thank Prof. N. Tomczak-Jaegermann for introducing me to this topic and for valuable suggestions on the text. Also, I thank Prof. G. Schechtman for a fruitful conversation.
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Tikhomirov, K.E. (2014). The Randomized Dvoretzky’s Theorem in \(l_{\infty }^{n}\) and the χ-Distribution. 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_31
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DOI: https://doi.org/10.1007/978-3-319-09477-9_31
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