Skip to main content Accessibility help
×
×
Home

A Short Proof of Paouris' Inequality

  • Radosław Adamczak (a1), Rafał Latała (a1), Alexander E. Litvak (a2), Krzysztof Oleszkiewicz (a1), Alain Pajor (a3) and Nicole Tomczak-Jaegermann (a2)...

Abstract

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $\left| X \right|$ of an isotropic log-concave random vector $X\,\in \,{{\mathbb{R}}^{n}},$ stating that for every $t\,\ge \,1$ ,

$$\mathbb{P}\left( \left| X \right|\,\ge \,ct\sqrt{n} \right)\,\le \,\exp (-t\sqrt{n}).$$

More precisely we show that for any log-concave random vector $X$ and any $p\,\ge \,1$ ,

$${{(\mathbb{E}{{\left| X \right|}^{p}})}^{1/p}}\,\sim \,\mathbb{E}\left| X \right|\,+\,\underset{z\in {{S}^{n-1}}}{\mathop{\sup }}\,\,{{(\mathbb{E}{{\left| \left\langle z,\,X \right\rangle \right|}^{p}})}^{1/p}}.$$

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A Short Proof of Paouris' Inequality
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      A Short Proof of Paouris' Inequality
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      A Short Proof of Paouris' Inequality
      Available formats
      ×

Copyright

References

Hide All
[1] Adamczak, R., Guédon, O., Latała, R., Litvak, A. E., Oleszkiewicz, K., Pajor, A., and Tomczak-Jaegermann, N., Moment estimates for convex measures. arxiv:1207.6618
[2] Adamczak, R., Latała, R., Litvak, A. E., Pajor, A., and Tomczak-Jaegermann, N., Tail estimates fornorms of sums of log-concave random vectors. arxiv:1107.4070.
[3] Adamczak, R., Geometry of log-concave ensembles of random matrices and approximate reconstruction. C. R. Math. Acad. Sci. Paris 349 (2011), no. 1314, 783786. http://dx.doi.org/10.1016/j.crma.2011.06.025
[4] Barlow, R. E., Marshall, A.W., and Proschan, F., Properties of probability distributions with monotonehazard rate. Ann. Math. Statist. 34 (1963), 375389. http://dx.doi.org/10.1214/aoms/1177704147
[5] Borell, C., Convex measures on locally convex spaces. Ark. Mat. 12 (1974), 239252. http://dx.doi.org/10.1007/BF02384761
[6] Borell, C., Convex set functions in d-space. Period. Math. Hungar. 6 (1975), no. 2, 111136. http://dx.doi.org/10.1007/BF02018814
[7] Borell, C., The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 (1975), no. 2, 207216. http://dx.doi.org/10.1007/BF01425510
[8] Davidovic, Ju. S., Korenbljum, B. I., and Hacet, B. I., A certain property of logarithmically concavefunctions. Soviet Math. Dokl. 10 (1969), 447480; translation from Dokl. Akad. Nauk SSSR 185 (1969), 12151218.
[9] Gordon, Y., Some inequalities for Gaussian processes and applications. Israel J. Math. 50 (1985), no. 4, 265289. http://dx.doi.org/10.1007/BF02759761
[10] Kwapień, S. , A remark on the median and the expectation of convex functions of Gaussian vectors. In: Probability in Banach spaces, 9 (Sandjberg, 1993), Progr. Probab., 35, Birkhäuser Boston, Boston, MA, 1994, pp. 271272.
[11] Kwapień, S., Latała, R., and Oleszkiewicz, K., Comparison of moments of sums of independent randomvariables and differential inequalities. J. Funct. Anal. 136 (1996), no. 1, 258268. http://dx.doi.org/10.1006/jfan.1996.0030
[12] Lifshits, M. A., Gaussian random functions. Mathematics and its Applications, 322, Kluwer Academic Publishers, Dordrecht, 1995.
[13] Litvak, A. E., Milman, V. D., and Schechtman, G., Averages of norms and quasi-norms. Math. Ann. 312 (1998), no. 1, 95124. http://dx.doi.org/10.1007/s002080050213
[14] Paouris, G., Concentration of mass on convex bodies. Geom. Funct. Anal. 16 (2006), no. 5, 10211049. http://dx.doi.org/10.1007/s00039-006-0584-5
[15] Sudakov, V. N. and Cirel’son, B. S., Extremal properties of half-spaces for spherically invariantmeasures. J. Sov. Math. 9 (1978), 918; translation from Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 41 (1974), 1424, 165.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed