Uniform almost sub-gaussian estimates for linear functionals on convex sets

B. Klartag

Algebra i analiz, Tome 19 (2007) no. 1, pp. 109-148.

Résumé

A well-known consequence of the Brunn–Minkowski inequality says that the distribution of a linear functional on a convex set has a uniformly subexponential tail. That is, for any dimension $n$, any convex set $K\subset\mathbb{R}^n$ of volume one, and any linear functional $\varphi\colon\mathbb{R}^n\to\mathbb{R}$, we have $$ \operatorname{Vol}_n(\{x\in K;|\varphi(x)|>t\|\varphi\|_{L_1(K)}\})\le e^{-ct}\quad \text{for all}\quad t>1, $$ where $\|\varphi\|_{L_1(K)}=\int_K|\varphi(x)|\,dx$ and $c>0$ is a universal constant. In this paper, it is proved that for any dimension $n$ and a convex set $K\subset\mathbb{R}^n$ of volume one, there exists a nonzero linear functional $\varphi\colon\mathbb{R}^n\to\mathbb{R}$ such that $$ \operatorname{Vol}_n(\{x\in K;|\varphi(x)|>t\|\varphi\|_{L_1(K)}\})\le e^{-c\frac{t^2}{\log^5 (t+1)}} \quad \text{for all}\quad t>1, $$ where $c>0$ is a universal constant.

Bibliographie

[1] Ball K., Isometric problems in $\ell_p$ and sections of convex sets, Ph. D. Dissertation, Trinity College, Cambridge, 1986

[2] Barthe F., Guédon O., Mendelson S., Naor A., “A probabilistic approach to the geometry of the $l_p^n$-ball”, Ann. Probab., 33 (2005), 480–513 | DOI | MR | Zbl

[3] Bobkov S. G., “On concentration of distributions of random weighted sums,”, Ann. Probab., 31:1 (2003), 195–215 | DOI | MR | Zbl

[4] Bobkov S. G., Nazarov F. L., “On convex bodies and log-concave probability measures with unconditional basis”, Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1807, Springer, Berlin, 2003, 53–69 | MR | Zbl

[5] Bobkov S. G., Nazarov F. L., “Large deviations of typical linear functionals on a convex body with unconditional basis”, Stochastic Inequalities and Applications, Progr. Probab., 56, Birkhäuser, Basel, 2003, 3–13 | MR | Zbl

[6] Borell C., “Convex measures on locally convex spaces”, Ark. Mat., 12 (1974), 239–252 | DOI | MR | Zbl

[7] Borell C., “Convex set functions in $d$-space”, Period. Math. Hungar., 6:2 (1975), 111–136 | DOI | MR

[8] Bourgain J., “On the distribution of polynomials on high-dimensional convex sets”, Geometric Aspects of Functional Analysis (1989–1990), Lecture Notes in Math., 1469, Springer, Berlin, 1991, 127–137 | MR

[9] Bourgain J., “On the isotropy-constant problem for “PSI-2”-bodies”, Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1807, Springer, Berlin, 2003, 114–121 | MR | Zbl

[10] Bourgain J., Milman V., “New volume ratio properties for convex symmetric bodies inn $R^n$”, Invent. Math., 88:2 (1987), 319–340 | DOI | MR | Zbl

[11] Brascamp H., Lieb E., “On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation”, J. Funct. Anal., 22:4 (1976), 366–389 | DOI | MR | Zbl

[12] Soviet Math. Dokl., 10 (1969), 477–480 | Zbl

[13] Dinghas A., “Über eine Klasse superadditiver Mengenfunktionale von Brunn–Minkowski–Lusternik-schem Typus”, Math. Z., 68 (1957), 111–125 | DOI | MR | Zbl

[14] Dinghas A., “Über zwei allgemeine Sätze von Brunn–Minkowski–Lusternikschem Typus”, Norske Vid. Selsk. Forh., Trondheim, 28 (1955), 182–185, (1956) | MR

[15] Grünbaum B., “Partitions of mass-distributions and of convex bodies by hyperplanes”, Pacific J. Math., 10 (1960), 1257–1261 | MR | Zbl

[16] Henstock R., Macbeath A. M., “On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik”, Proc. London Math. Soc. (3), 3 (1953), 182–194 | DOI | MR | Zbl

[17] Klartag B., “A geometric inequality and a low $M$-estimate”, Proc. Amer. Math. Soc., 132:9 (2004), 2619–2628 | DOI | MR | Zbl

[18] Klartag B., “An isomorphic version of the slicing problem”, J. Funct. Anal., 218 (2005), 372–394 | DOI | MR | Zbl

[19] Klartag B., “Marginals of geometric inequalities”, Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1910, Springer, Berlin, 2007, 133–166 | MR | Zbl

[20] Klartag B., “On convex perturbations with a bounded isotropic constant”, Geom. Funct. Anal., 16:6 (2006), 1274–1290 | DOI | MR | Zbl

[21] Klartag B., Milman V., “Geometry of log-concave functions and measures”, Geom. Dedicata, 112 (2005), 169–182 | DOI | MR | Zbl

[22] Leindler L., “On a certain converse of Hölder's inequality”, Linear Operators and Approximation, Proc. Conf., Oberwolfach, 1971, Internat. Ser. Numer. Math., 20, Birkhäuser, Basel, 1972, 182–184 | MR

[23] Lekkerkerker C. G., “A property of logarithmic concave functions, I, II”, Nederl. Akad. Wetensch. Proc. Ser. A, 56 (1953), 505–521 | MR | Zbl

[24] Meyer M., Pajor A., “On Santaló's inequality”, Geometric Aspects of Functional Analysis (1987–1988), Lecture Notes in Math., 1376, Springer, Berlin, 1989, 261–263 | MR

[25] Milman V., “Randomness and pattern in convex geometric analysis”, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math., Extra Vol. II, 1998, 665–677 | MR | Zbl

[26] Milman V., Pajor A., “Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space”, Geometric Aspects of Functional Analysis, Israel Seminar (1987–88), Lecture Notes in Math., 1376, Springer, Berlin, 1989, 64–104 | MR

[27] Milman V., Schechtman G., Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Math., 1200, Springer-Verlag, Berlin, 1986 | MR | Zbl

[28] Paouris G., “$\Psi_2$-estimates for linear functionals on zonoids”, Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1807, Springer, Berlin, 2003, 211–222 | MR | Zbl

[29] Paouris G., “On the $\Psi_2$ behavior of linear functionals on isotropic convex bodies”, Studia Math., 168:3 (2005), 285–299 | DOI | MR | Zbl

[30] Paouris G., “Concentration of mass on isotropic convex bodies”, C. R. Math. Acad. Sci. Paris, 342:3 (2006), 179–182 | MR | Zbl

[31] Pisier G., The volume of convex bodies and Banach space geometry, Cambridge Tracts in Math., 94, Cambridge Univ. Press, Cambridge, 1989 | MR | Zbl

[32] Prékopa A., “Logarithmic concave measures with application to stochastic programming”, Acta Sci. Math. (Szeged), 32 (1971), 301–316 | MR | Zbl

[33] Prékopa A., “On logarithmic concave measures and functions”, Acta Sci. Math. (Szeged), 34 (1973), 335–343 | MR | Zbl

[34] Santaló L. A., “An affine invariant for convex bodies of $n$-dimensional space”, Portugaliae Math., 8 (1949), 155–161 | MR | Zbl

[35] Schneider R., Convex bodies: the Brunn–Minkowski theory, Encyclopedia Math. Appl., 44, Cambridge Univ. Press, Cambridge, 1993 | MR | Zbl