Estimates for moments of general measures on convex bodies
HTML articles powered by AMS MathViewer
- by Sergey Bobkov, Bo’az Klartag and Alexander Koldobsky PDF
- Proc. Amer. Math. Soc. 146 (2018), 4879-4888 Request permission
Abstract:
For $p\ge 1$, $n\in \mathbb {N}$, and an origin-symmetric convex body $K$ in $\mathbb {R}^n,$ let \begin{equation*} d_\textrm {{ovr}}(K,L_p^n) = \inf \left \{ \Big (\frac {|D|}{|K|}\Big )^{1/n}: K \subseteq D,\ D\in L_p^n \right \} \end{equation*} be the outer volume ratio distance from $K$ to the class $L_p^n$ of the unit balls of $n$-dimensional subspaces of $L_p.$ We prove that there exists an absolute constant $c>0$ such that \begin{equation*}\frac {c\sqrt {n}}{\sqrt {p\log \log n}}\le \sup _K d_\textrm {{ovr}}(K,L_p^n)\le \sqrt {n}. \end{equation*} This result follows from a new slicing inequality for arbitrary measures, in the spirit of the slicing problem of Bourgain. Namely, there exists an absolute constant $C>0$ so that for any $p\ge 1,$ any $n\in \mathbb {N}$, any compact set $K \subseteq \mathbb {R}^n$ of positive volume, and any Borel measurable function $f\ge 0$ on $K$, \begin{equation*}\int _K f(x) dx \le C \sqrt {p}\ d_\textrm {ovr}(K,L_p^n)\ |K|^{1/n} \sup _{H} \int _{K\cap H} f(x) dx, \end{equation*} where the supremum is taken over all affine hyperplanes $H$ in $\mathbb {R}^n$. Combining the above display with a recent counterexample for the slicing problem with arbitrary measures from the work of the second and third authors [J. Funct. Anal. 274 (2018), pp. 2089–2112], we get the lower estimate from the first display.
In turn, the second inequality follows from an estimate for the $p$-th absolute moments of the function $f$ \begin{equation*} \min _{\xi \in S^{n-1}} \int _K |(x,\xi )|^p f(x)\ dx \le (Cp)^{p/2} d^p_\textrm {{ovr}}(K,L_p^n)\ |K|^{p/n} \int _K f(x) dx. \end{equation*} Finally, we prove a result of the Busemann-Petty type for these moments.
References
- Keith Ball, Normed spaces with a weak-Gordon-Lewis property, Functional analysis (Austin, TX, 1987/1989) Lecture Notes in Math., vol. 1470, Springer, Berlin, 1991, pp. 36–47. MR 1126735, DOI 10.1007/BFb0090210
- J. Bourgain, On high-dimensional maximal functions associated to convex bodies, Amer. J. Math. 108 (1986), no. 6, 1467–1476. MR 868898, DOI 10.2307/2374532
- J. Bourgain, Geometry of Banach spaces and harmonic analysis, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 871–878. MR 934289
- J. Bourgain, On the distribution of polynomials on high-dimensional convex sets, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 127–137. MR 1122617, DOI 10.1007/BFb0089219
- Giorgos Chasapis, Apostolos Giannopoulos, and Dimitris-Marios Liakopoulos, Estimates for measures of lower dimensional sections of convex bodies, Adv. Math. 306 (2017), 880–904. MR 3581320, DOI 10.1016/j.aim.2016.10.035
- Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948, pp. 187–204. MR 0030135
- B. Klartag, An isomorphic version of the slicing problem, J. Funct. Anal. 218 (2005), no. 2, 372–394. MR 2108116, DOI 10.1016/j.jfa.2004.05.003
- B. Klartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. 16 (2006), no. 6, 1274–1290. MR 2276540, DOI 10.1007/s00039-006-0588-1
- Bo’az Klartag and Alexander Koldobsky, An example related to the slicing inequality for general measures, J. Funct. Anal. 274 (2018), no. 7, 2089–2112. MR 3762096, DOI 10.1016/j.jfa.2017.08.025
- Alexander Koldobsky, Intersection bodies, positive definite distributions, and the Busemann-Petty problem, Amer. J. Math. 120 (1998), no. 4, 827–840. MR 1637955
- Alexander Koldobsky, Fourier analysis in convex geometry, Mathematical Surveys and Monographs, vol. 116, American Mathematical Society, Providence, RI, 2005. MR 2132704, DOI 10.1090/surv/116
- Alexander Koldobsky, A hyperplane inequality for measures of convex bodies in $\Bbb R^n$, $n\leq 4$, Discrete Comput. Geom. 47 (2012), no. 3, 538–547. MR 2891246, DOI 10.1007/s00454-011-9362-8
- Alexander Koldobsky, A $\sqrt {n}$ estimate for measures of hyperplane sections of convex bodies, Adv. Math. 254 (2014), 33–40. MR 3161089, DOI 10.1016/j.aim.2013.12.029
- Alexander Koldobsky, Slicing inequalities for measures of convex bodies, Adv. Math. 283 (2015), 473–488. MR 3383809, DOI 10.1016/j.aim.2015.07.019
- Alexander Koldobsky and Alain Pajor, A remark on measures of sections of $L_p$-balls, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2169, Springer, Cham, 2017, pp. 213–220. MR 3645124
- Alexander Koldobsky and Artem Zvavitch, An isomorphic version of the Busemann-Petty problem for arbitrary measures, Geom. Dedicata 174 (2015), 261–277. MR 3303052, DOI 10.1007/s10711-014-0016-x
- Erwin Lutwak, Intersection bodies and dual mixed volumes, Adv. in Math. 71 (1988), no. 2, 232–261. MR 963487, DOI 10.1016/0001-8708(88)90077-1
- Emanuel Milman, Dual mixed volumes and the slicing problem, Adv. Math. 207 (2006), no. 2, 566–598. MR 2271017, DOI 10.1016/j.aim.2005.09.008
- A. Zvavitch, The Busemann-Petty problem for arbitrary measures, Math. Ann. 331 (2005), no. 4, 867–887. MR 2148800, DOI 10.1007/s00208-004-0611-5
Additional Information
- Sergey Bobkov
- Affiliation: Department of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, Minnesota 55455
- MR Author ID: 197155
- Email: bobkov@math.umn.edu
- Bo’az Klartag
- Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot 76100 Israel – and – School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978
- MR Author ID: 671208
- Email: boaz.klartag@weizmann.ac.il
- Alexander Koldobsky
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 104225
- Email: koldobskiya@missouri.edu
- Received by editor(s): December 16, 2017
- Received by editor(s) in revised form: February 1, 2018
- Published electronically: July 23, 2018
- Additional Notes: This material is based upon work supported by the U. S. National Science Foundation under Grant DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. The first- and third-named authors were supported in part by the NSF Grants DMS-1612961 and DMS-1700036. The second-named author was supported in part by a European Research Council (ERC) grant.
- Communicated by: Thomas Schlumprecht
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4879-4888
- MSC (2010): Primary 52A20; Secondary 46B07
- DOI: https://doi.org/10.1090/proc/14119
- MathSciNet review: 3856154