An isoperimetric inequality on the ${\ell }_p$ balls

Sodin, Sasha

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 362-373.

Nous prouvons une inégalité isopérimétrique pour la mesure uniforme $V_{p,n}$ sur la boule unité de ${\ell }_p^n(1\le p\le 2)$. Si $V_{p,n}\left(A\right)=a$, alors $V_{p,n}^{+}\left(A\right)\ge c{n}^{1/p}\tilde{a}{log}^{1-1/p}1/\tilde{a}$, où $V_{p,n}^{+}$ est la mesure de surface associée à $V_{p,n},\tilde{a}=min(a,1-a)$ et $c$ est une constante absolue. En particulier, les boules unités de ${\ell }_p^n$ vérifient la conjecture de Kannan-Lovász-Simonovits (Discrete Comput. Geom. 13 (1995)) sur la constante de Cheeger d'un corps convexe isotrope. La démonstration s'appuie sur les inégalités isopérimétriques de Bobkov (Ann. Probab. 27 (1999)) et de Barthe-Cattiaux-Roberto (Rev. Math. Iberoamericana 22 (2006)), et utilise la représentation de $V_{p,n}$ établie par Barthe-Guédon-Mendelson-Naor (Ann. Probab. 33 (2005)) ainsi qu'un argument de découpage.

The normalised volume measure on the ${\ell }_p^n$ unit ball $(1\le p\le 2)$ satisfies the following isoperimetric inequality: the boundary measure of a set of measure $a$ is at least $c{n}^{1/p}\tilde{a}{log}^{1-1/p}(1/\tilde{a})$, where $\tilde{a}=min(a,1-a)$.

[1] F. J. Almgren. Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Amer. Math. Soc. 4 (1976) 165. | MR | Zbl

[2] F. Barthe. Log-concave and spherical models in isoperimetry. Geom. Funct. Anal. 12 (2002) 32-55. | MR | Zbl

[3] F. Barthe, P. Cattiaux and C. Roberto. Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Math. Iberoamericana 22 (2005) 993-1067. | MR | Zbl

[4] F. Barthe, O. Guédon, S. Mendelson and A. Naor. A probabilistic approach to the geometry of the ℓnp-ball. Ann. Probab. 33 (2005) 480-513. | MR | Zbl

[5] F. Barthe and C. Roberto. Sobolev inequalities for probability measures on the real line. Studia Math. 159 (2003) 481-497. | MR | Zbl

[6] S. G. Bobkov. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 (1999) 1903-1921. | MR | Zbl

[7] S. G. Bobkov. Spectral gap and concentration for some spherically symmetric probability measures. Geometric Aspects of Functional Analysis (Notes of GAFA Seminar) 37-43. Lecture Notes in Math. 1807. Springer, Berlin, 2003. | MR | Zbl

[8] S. G. Bobkov. An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab. 25 (1997) 206-214. | MR | Zbl

[9] S. G. Bobkov. On isoperimetric constants for log-concave probability distributions. Lecture Notes in Math. 1910 (2007) 81-88. | MR | Zbl

[10] S. Bobkov. Extremal properties of half-spaces for log-concave distributions. Ann. Probab. 24 (1996) 35-48. | MR | Zbl

[11] S. G. Bobkov and C. Houdré. Isoperimetric constants for product probability measures. Ann. Probab. 25 (1997) 184-205. | MR | Zbl

[12] S. G. Bobkov and B. Zegarlinski. Entropy bounds and isoperimetry. Mem. Amer. Math. Soc. 176 (2005) 829. | MR | Zbl

[13] J. Bokowski and E. Sperner. Zerlegung konvexer Körper durch minimale Trennflächen. J. Reine Angew. Math. 311/312 (1979) 80-100. | MR | Zbl

[14] C. Borell. The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 (1975) 207-216. | MR | Zbl

[15] J. Bourgain. On the distribution of polynomials on high-dimensional convex sets. Geometric aspects of functional analysis (Notes of GAFA Seminar) 127-137. Lecture Notes in Math. 1469. Springer, Berlin, 1991. | MR | Zbl

[16] Y. D. Burago and V. G. Maz'Ja. Certain questions of potential theory and function theory for regions with irregular boundaries. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 3 (1967) 152. (English translation: Potential theory and function theory for irregular regions. Seminars in Mathematics. V. A. Steklov Mathematical Institute, Leningrad, Vol. 3, Consultants Bureau, New York, 1969) | MR | Zbl

[17] H. Federer and W. H. Fleming. Normal and integral currents. Ann. of Math. (2) 72 (1960) 458-520. | MR | Zbl

[18] B. Klartag. On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16 (2006) 1274-1290. | MR | Zbl

[19] R. Kannan, L. Lovász and M. Simonovits. Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13 (1995) 541-559. | MR | Zbl

[20] M. Ledoux. Spectral gap, logarithmic Sobolev constant, and geometric bounds. In Surveys in Differential Geometry. Vol. IX 219-240. Int. Press, Somerville, MA, 2004. | MR | Zbl

[21] R. Latała and K. Oleszkiewicz. Between Sobolev and Poincaré. Geometric Aspects of Functional Analysis (Notes of GAFA Seminar) 147-168. Lecture Notes in Math. 1745. Springer, Berlin, 2000. | MR | Zbl

[22] V. G. Maz'Ja. Classes of domains and imbedding theorems for function spaces. Dokl. Akad. Nauk SSSR 133 (1960) 527-530. (Russian). Translated as Soviet Math. Dokl. 1 (1960) 882-885. | MR | Zbl

[23] G. Paouris. Concentration of mass on isotropic convex bodies. C. R. Math. Acad. Sci. Paris 342 (2006) 179-182. | MR | Zbl

[24] G. Schechtman and J. Zinn. On the volume of the intersection of two Lnp balls. Proc. Amer. Math. Soc. 110 (1990) 217-224. | MR | Zbl

[25] G. Schechtman and J. Zinn. Concentration on the ℓnp ball. Geometric Aspects of Functional Analysis (Notes of GAFA Seminar) 245-256. Lecture Notes in Math. 1745. Springer, Berlin, 2000. | MR | Zbl

[26] V. N. Sudakov and B. S. Cirel'Son. Extremal properties of half-spaces for spherically invariant measures. Problems in the theory of probability distributions, II. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974) 14-24, 165. (Russian). | MR | Zbl