Abstract
We show that there exists a sequence \(\varepsilon_n\searrow0\) for which the following holds: Let K⊂ℝn be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝn, t0∈ℝ and σ>0 such that
where the supremum runs over all measurable sets A⊂ℝ, and where 〈·,·〉 denotes the usual scalar product in ℝn. Furthermore, under the additional assumptions that the expectation of X is zero and that the covariance matrix of X is the identity matrix, we may assert that most unit vectors θ satisfy (*), with t0=0 and σ=1. Corresponding principles also hold for multi-dimensional marginal distributions of convex sets.
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Anttila, M., Ball, K., Perissinaki, I.: The central limit problem for convex bodies. Trans. Am. Math. Soc. 355(12), 4723–4735 (2003)
Ball, K.: Logarithmically concave functions and sections of convex sets in ℝn. Stud. Math. 88(1), 69–84 (1988)
Bastero, J., Bernués, J.: Asymptotic behavior of averages of k-dimensional marginals of measures on ℝn. Preprint. Available at http://www.unizar.es/galdeano/preprints/2005/preprint34.pdf
Bobkov, S.G.: On concentration of distributions of random weighted sums. Ann. Probab. 31(1), 195–215 (2003)
Bobkov, S.G., Koldobsky, A.: On the central limit property of convex bodies. In: Geometric Aspects of Functional Analysis, Israel Seminar (2001–02). Lect. Notes Math., vol. 1807, pp. 44–52. Springer, Berlin (2003)
Borell, C.: Convex measures on locally convex spaces. Ark. Mat. 12, 239–252 (1974)
Brehm, U., Voigt, J.: Asymptotics of cross sections for convex bodies. Beitr. Algebra Geom. 41(2), 437–454 (2000)
Brehm, U., Hinow, P., Vogt, H., Voigt, J.: Moment inequalities and central limit properties of isotropic convex bodies. Math. Z. 240(1), 37–51 (2002)
Dasgupta, S., Gupta, A.: An elementary proof of a theorem of Johnson and Lindenstrauss. Random Struct. Algorithms 22(1), 60–65 (2003)
Davidovič, J.S., Korenbljum, B.I., Hacet, B.I.: A certain property of logarithmically concave functions. Dokl., Akad. Nauk Azerb. 185, 1215–1218 (1969) (Russian). English translation in Sov. Math. Dokl. 10, 477–480 (1969)
Diaconis, P., Freedman, D.: Asymptotics of graphical projection pursuit. Ann. Stat. 12(3), 793–815 (1984)
Diaconis, P., Freedman, D.: A dozen de Finetti-style results in search of a theory. Ann. Inst. Henri Poincaré, Probab. Stat. 23(2), 397–423 (1987)
Feller, W.: An Introduction to Probability Theory and its Applications, Volume II. John Wiley & Sons, Inc., New York, London, Sydney, (1971)
Fradelizi, M.: Sections of convex bodies through their centroid. Arch. Math. 69(6), 515–522 (1997)
Giannopoulos, A.A., Milman, V.D.: Euclidean structure in finite dimensional normed spaces. In: Handbook of the Geometry of Banach Spaces, Vol. I, pp. 707–779. North-Holland, Amsterdam (2001)
Gordon, Y.: Gaussian processes and almost spherical sections of convex bodies. Ann. Probab. 16(1), 180–188 (1988)
Gordon, Y.: On Milman’s inequality and random subspaces which escape through a mesh in ℝn. In: Geometric Aspects of Functional Analysis, Israel Seminar (1986–87). Lect. Notes Math., vol. 1317, pp. 84–106. Springer, Berlin (1988)
Gromov, M.: Dimension, nonlinear spectra and width. In: Geometric Aspects of Functional Analysis, Israel Seminar (1986–87). Lect. Notes Math., vol. 1317, pp. 132–184. Springer, Berlin (1988)
Grünbaum, B.: Partitions of mass-distributions and of convex bodies by hyperplanes. Pac. J. Math. 10, 1257–1261 (1960)
Hensley, D.: Slicing convex bodies – bounds for slice area in terms of the body’s covariance. Proc. Am. Math. Soc. 79(4), 619–625 (1980)
Johnson, W.B., Lindenstrauss, J.: Extensions of Lipschitz mappings into a Hilbert space. In: Conference in Modern Analysis and Probability (New Haven, Conn., 1982). Contemp. Math. vol. 26, pp. 189–206. Am. Math. Soc. Providence, RI (1984)
Klartag, B., Milman, V.D.: Geometry of log-concave functions and measures. Geom. Dedicata 112, 169–182 (2005)
Klartag, B.: Uniform almost sub-gaussian estimates for linear functionals on convex sets. To appear in Algebra Anal. (St. Petersb. Math. J.)
Koldobsky, A., Lifshits, M.: Average volume of sections of star bodies. In: Geometric Aspects of Functional Analysis, Israel Seminar (1996–00). Lect. Notes Math., vol. 1745, pp. 119–146. Springer, Berlin (2000)
Ledoux, M.: Spectral gap, logarithmic Sobolev constant, and geometric bounds. Appears in: Eigenvalues of Laplacians and Other Geometric Operators. Surveys in Differential Geometry, vol. 9, pp. 219–240. Somerville, MA (2004)
Leindler, L.: On a certain converse of Hölder’s inequality. Linear operators and approximation. Proc. Conf., Oberwolfach, (1971). Int. Ser. Numer. Math., vol. 20, pp. 182–184. Birkhäuser, Basel (1972)
Lekkerkerker, C.G.: A property of Logarithmic concave functions I+II. Indag. Math., New Ser. 15, 505–521 (1953) (also known as Nederl. Akad. Wetensch. Proc. Ser. A. 56)
Lindenstrauss, J.: Almost spherical sections; their existence and their applications. Jahresber. Dtsch. Math.-Ver. 39–61 (1992)
Lovász, L., Vempala, S.: The geometry of logconcave functions and sampling algorithms. Random Struct. Algorithms (published online)
Meckes, E.S., Meckes, M.W.: The central limit problem for random vectors with symmetries. Preprint.
Milman, E.: On gaussian marginals of uniformly convex bodies. Preprint. Available at http://arxiv.org/abs/math.FA/0604595
Milman, V.D.: A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies. Funkts. Anal. Prilozh. 5(4), 28–37 (1971) (Russian) English translation in Funct. Anal. Appl. 5, 288–295 (1971)
Milman, V.D.: Dvoretzky’s theorem – thirty years later. Geom. Funct. Anal. 2(4), 455–479 (1992)
Milman, V.D., Pajor, A.: Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space. In: Geometric Aspects of Functional Analysis, Israel Seminar (1987–88). Lect. Notes Math., vol. 1376, pp. 64–104. Springer, Berlin (1989)
Milman, V.D., Schechtman, G.: Asymptotic Theory of Finite-Dimensional Normed Spaces. Lect. Notes Math., vol. 1200. Springer, Berlin (1986)
Naor, A., Romik, D.: Projecting the surface measure of the sphere of ℓ p n. Ann. Inst. Henri Poincaré, Probab. Stat. 39(2), 241–261 (2003)
Paouris, G.: Concentration of mass and central limit properties of isotropic convex bodies. Proc. Am. Math. Soc. 133, 565–575 (2005)
Paouris, G.: On the ψ2-behaviour of linear functionals on isotropic convex bodies. Stud. Math. 168(3), 285–299 (2005)
Paouris, G.: Concentration of mass on isotropic convex bodies. C. R. Math. Acad. Sci., Paris 342(3), 179–182 (2006)
Paouris, G.: Concentration of mass in convex bodies. Geom. Funct. Anal. 16(5), 1021–1049 (2006)
Pisier, G.: The volume of convex bodies and Banach space geometry. In: Cambridge Tracts in Mathematics, vol. 94. Cambridge University Press, Cambridge (1989)
Prékopa, A.: Logarithmic concave measures with application to stochastic programming. Acta Sci. Math. 32, 301–316 (1971)
Prékopa, A.: On logarithmic concave measures and functions. Acta Sci. Math. 34, 335–343 (1973)
Romik, D.: Randomized Central Limit Theorems – Probabilisitic and Geometric Aspects. PhD dissertation. Tel-Aviv University (2001)
Schechtman, G.: A remark concerning the dependence on ε in Dvoretzky’s theorem. In: Geometric Aspects of Functional Analysis, Israel Seminar (1987–88). Lect. Notes Math., vol. 1376, pp. 274–277. Springer, Berlin (1989)
Schechtman, G.: Two observations regarding embedding subsets of Euclidean spaces in normed spaces. Geom. Funct. Anal. 200, 125–135 (2006)
Schoenberg, I.J.: On Pólya frequency functions. I. The totally positive functions and their Laplace transforms. J. Anal. Math. 1, 331–374 (1951)
Sodin, S.: Tail-sensitive gaussian asymptotics for marginals of concentrated measures in high dimension. To appear in: Geometric Aspects of Functional Analysis, Israel Seminar. Lect. Notes Math. Available at http://arxiv.org/abs/math.MG/0501382
Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, no. 32. Princeton University Press, Princeton, NJ (1971)
Stroock, D.W.: Probability Theory, an Analytic View. Cambridge University Press, Cambridge (1993)
Sudakov, V.N.: Typical distributions of linear functionals in finite-dimensional spaces of high-dimension. Dokl., Akad. Nauk Azerb. 243(6), 1402–1405 (1978) (Russian) English translation in Sov. Math. Dokl. 19, 1578–1582 (1978)
von Weizsäcker, H.: Sudakov’s typical marginals, random linear functionals and a conditional central limit theorem. Probab. Theory Relat. Fields 107(3), 313–324 (1997)
Wojtaszczyk, J.O.: The square negative correlation property for generalized orlicz balls. To appear in: Geometric Aspects of Functional Analysis, Israel Seminar. Lect. Notes Math.
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Klartag, B. A central limit theorem for convex sets. Invent. math. 168, 91–131 (2007). https://doi.org/10.1007/s00222-006-0028-8
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DOI: https://doi.org/10.1007/s00222-006-0028-8