Geodesic
Kahane-Khinchin’s Inequality for Quasi-Norms
Canadian mathematical bulletin, Tome 43 (2000) no. 3, pp. 368-379

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DOI

We extend the recent results of R. Latała and O. Guédon about equivalence of ${{L}_{q}}$ -norms of logconcave random variables (Kahane-Khinchin’s inequality) to the quasi-convex case. We construct examples of quasi-convex bodies ${{K}_{n\,}}\subset {{\mathbb{R}}^{n}}$ which demonstrate that this equivalence fails for uniformly distributed vector on ${{K}_{n}}$ (recall that the uniformly distributed vector on a convex body is logconcave). Our examples also show the lack of the exponential decay of the “tail” volume (for convex bodies such decay was proved by M. Gromov and V. Milman).
DOI : 10.4153/CMB-2000-044-4
Mots-clés : 46B09, 52A30, 60B11 
Litvak, A. E. Kahane-Khinchin’s Inequality for Quasi-Norms. Canadian mathematical bulletin, Tome 43 (2000) no. 3, pp. 368-379. doi: 10.4153/CMB-2000-044-4
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     title = {Kahane-Khinchin{\textquoteright}s {Inequality} for {Quasi-Norms}},
     journal = {Canadian mathematical bulletin},
     pages = {368--379},
     year = {2000},
     volume = {43},
     number = {3},
     doi = {10.4153/CMB-2000-044-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2000-044-4/}
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