## Slicing inequalities for subspaces of $L_p$

HTML articles powered by AMS MathViewer

- by Alexander Koldobsky PDF
- Proc. Amer. Math. Soc.
**144**(2016), 787-795 Request permission

## Abstract:

We prove slicing inequalities for measures of the unit balls of subspaces of $L_p$, $-\infty <p<\infty .$ For example, for every $k\in \mathbb {N}$ there exists a constant $C(k)$ such that for every $n\in \mathbb {N},\ k<n$, every convex $k$-intersection body (unit ball of a normed subspace of $L_{-k})$ $L$ in $\mathbb {R}^n$ and every measure $\mu$ with non-negative even continuous density in $\mathbb {R}^n,$ \[ \mu (L)\ \le \ C(k) \max _{\xi \in S^{n-1}} \mu (L\cap \xi ^\bot )\ |L|^{1/n} \ ,\] where $\xi ^\bot$ is the central hyperplane in $\mathbb {R}^n$ perpendicular to $\xi ,$ and $|L|$ is the volume of $L.$ This and other results are in the spirit of the hyperplane problem of Bourgain. The proofs are based on stability inequalities for intersection bodies and estimates for the Banach-Mazur distance from the unit ball of a subspace of $L_p$ to the class of intersection bodies.## References

- K. Ball,
*Isometric problems in $\ell _p$ and sections of convex sets*, Ph.D. dissertation, Trinity College, Cambridge (1986). - 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 - S. Brazitikos, A. Giannopoulos, P. Valettas, and B. Vritsiou,
*Geometry of isotropic log-concave measures*, preprint. - Richard J. Gardner,
*Geometric tomography*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, New York, 2006. MR**2251886**, DOI 10.1017/CBO9781107341029 - Eric Grinberg and Gaoyong Zhang,
*Convolutions, transforms, and convex bodies*, Proc. London Math. Soc. (3)**78**(1999), no. 1, 77–115. MR**1658156**, DOI 10.1112/S0024611599001653 - Marius Junge,
*Hyperplane conjecture for quotient spaces of $L_p$*, Forum Math.**6**(1994), no. 5, 617–635. MR**1295155**, DOI 10.1515/form.1994.6.617 - N. J. Kalton and A. Koldobsky,
*Intersection bodies and $L_p$-spaces*, Adv. Math.**196**(2005), no. 2, 257–275. MR**2166308**, DOI 10.1016/j.aim.2004.09.002 - N. J. Kalton, A. Koldobsky, V. Yaskin, and M. Yaskina,
*The geometry of $L_0$*, Canad. J. Math.**59**(2007), no. 5, 1029–1049. MR**2354401**, DOI 10.4153/CJM-2007-044-0 - N. J. Kalton and M. Zymonopoulou,
*Positive definite distributions and normed spaces*, Adv. Math.**227**(2011), no. 2, 986–1018. MR**2793030**, DOI 10.1016/j.aim.2011.02.019 - 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 - 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,
*Positive definite distributions and subspaces of $L_{-p}$ with applications to stable processes*, Canad. Math. Bull.**42**(1999), no. 3, 344–353. MR**1703694**, DOI 10.4153/CMB-1999-040-5 - A. Koldobsky,
*A generalization of the Busemann-Petty problem on sections of convex bodies*, Israel J. Math.**110**(1999), 75–91. MR**1750442**, DOI 10.1007/BF02808176 - A. Koldobsky,
*A functional analytic approach to intersection bodies*, Geom. Funct. Anal.**10**(2000), no. 6, 1507–1526. MR**1810751**, DOI 10.1007/PL00001659 - Alexander Koldobsky,
*Stability in the Busemann-Petty and Shephard problems*, Adv. Math.**228**(2011), no. 4, 2145–2161. MR**2836117**, DOI 10.1016/j.aim.2011.06.031 - 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 - A. Koldobsky,
*Estimates for measures of sections of convex bodies*, arXiv:1309.6485 - A. Koldobsky,
*Stability and separation in volume comparison problems*, Math. Model. Nat. Phenom.**8**(2013), no. 1, 156–169. MR**3022986**, DOI 10.1051/mmnp/20138111 - A. Koldobsky, A. Pajor, and V. Yaskin,
*Inequalities of the Kahane-Khinchin type and sections of $L_p$-balls*, Studia Math.**184**(2008), no. 3, 217–231. MR**2369140**, DOI 10.4064/sm184-3-2 - Alexander Koldobsky and Vladyslav Yaskin,
*The interface between convex geometry and harmonic analysis*, CBMS Regional Conference Series in Mathematics, vol. 108, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2008. MR**2365157** - D. R. Lewis,
*Finite dimensional subspaces of $L_{p}$*, Studia Math.**63**(1978), no. 2, 207–212. MR**511305**, DOI 10.4064/sm-63-2-207-212 - 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 - B. Maurey,
*Théoremes de factorization pour les operatèurs linéaires à valeurs dans les espaces $L_p$.*Astérisque**11**(1974), Société Math. de France. - 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 - Emanuel Milman,
*Generalized intersection bodies*, J. Funct. Anal.**240**(2006), no. 2, 530–567. MR**2261694**, DOI 10.1016/j.jfa.2006.04.004 - Emanuel Milman,
*Generalized intersection bodies are not equivalent*, Adv. Math.**217**(2008), no. 6, 2822–2840. MR**2397468**, DOI 10.1016/j.aim.2007.11.007 - V. D. Milman and A. Pajor,
*Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space*, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104. MR**1008717**, DOI 10.1007/BFb0090049 - E. M. Nikišin,
*Resonance theorems and superlinear operators*, Uspehi Mat. Nauk**25**(1970), no. 6(156), 129–191 (Russian). MR**0296584** - Gideon Schechtman and Artem Zvavitch,
*Embedding subspaces of $L_p$ into $l^N_p$, $0<p<1$*, Math. Nachr.**227**(2001), 133–142. MR**1840560**, DOI 10.1002/1522-2616(200107)227:1<133::AID-MANA133>3.3.CO;2- - Vladyslav Yaskin,
*On strict inclusions in hierarchies of convex bodies*, Proc. Amer. Math. Soc.**136**(2008), no. 9, 3281–3291. MR**2407094**, DOI 10.1090/S0002-9939-08-09424-0 - Vladyslav Yaskin,
*Counterexamples to convexity of $k$-intersection bodies*, Proc. Amer. Math. Soc.**142**(2014), no. 12, 4355–4363. MR**3267003**, DOI 10.1090/S0002-9939-2014-12254-4 - Gaoyong Zhang,
*Sections of convex bodies*, Amer. J. Math.**118**(1996), no. 2, 319–340. MR**1385280**

## Additional Information

**Alexander Koldobsky**- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 104225
- Email: koldobskiya@missouri.edu
- Received by editor(s): July 20, 2014
- Received by editor(s) in revised form: December 26, 2014
- Published electronically: May 28, 2015
- Additional Notes: This work was partially supported by the US National Science Foundation, grant DMS-1265155.
- Communicated by: Thomas Schlumprecht
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 787-795 - MSC (2010): Primary 52A20; Secondary 46B07
- DOI: https://doi.org/10.1090/proc12708
- MathSciNet review: 3430854