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Kahane-Khinchine type inequalities for negative exponent. (English) Zbl 0965.26011
Summary: We prove a concentration inequality for \(\delta\)-concave measures over \(\mathbb{R}^n\). Using this result, we study the moments of order \(q\) of a norm with respect to a \(\delta\)-concave measure over \(\mathbb{R}^n\). We obtain a lower bound for \(q\in ]-1,0]\) and an upper bound for \(q\in ]0,+\infty[\) in terms of the measure of the unit ball associated to the norm. This allows us to give Kahane-Khinchine type inequalities for a negative exponent.

26D15 Inequalities for sums, series and integrals
26A51 Convexity of real functions in one variable, generalizations
28A12 Contents, measures, outer measures, capacities
60E15 Inequalities; stochastic orderings
Full Text: DOI
[1] DOI: 10.1007/BF02767353 · Zbl 0654.46019 · doi:10.1007/BF02767353
[2] Milman, Springer Lecture Notes 1200 (1986)
[3] Milman, CR. Acad. Sci. Paris 308 pp 91– (1989)
[4] DOI: 10.1007/BF01425510 · Zbl 0292.60004 · doi:10.1007/BF01425510
[5] Latala, MSRI, Publications 34 pp 123– (1998)
[6] Kahane, Cambridge Studies in Advanced Math (1985)
[7] DOI: 10.1007/BF02018814 · doi:10.1007/BF02018814
[8] DOI: 10.1002/rsa.3240040402 · Zbl 0788.60087 · doi:10.1002/rsa.3240040402
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