Abstract
In the context of his work on maximal functions in the 1980s, Jean Bourgain came across the following geometric question: Is there c > 0 such that for any dimension n and any convex body \(K \subseteq \mathbb R^n\) of volume one, there exists a hyperplane H such that the (n − 1)-dimensional volume of K ∩ H is at least c? This innocent and seemingly obvious question (which remains unanswered!) has established a new direction in high-dimensional geometry. It has emerged as an “engine” that inspired the discovery of many deep results and unexpected connections. Here we provide a survey of these developments, including many of Bourgain’s results.
Dedicated to the memory of Jean Bourgain
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Adamczak, R., Latała, R., Litvak, A. E., Oleszkiewicz, K., Pajor, A., Tomczak-Jaegermann, N.: A short proof of Paouris’ inequality. Canad. Math. Bull. 57(1), 3–8 (2014)
Anttila, M., Ball, K., Perissinaki, I.: The central limit problem for convex bodies. Trans. Am. Math. Soc. 355(12), 4723–4735 (2003)
Autissier, P.: Un lemme matriciel effectif. Math. Z. 273(1–2), 355–361 (2013)
Ball, K.: Isometric problems in ℓ p and sections of convex sets. Ph.D. Thesis, Cambridge University, 1986
Ball, K.: Cube slicing in \(\mathbb R^n\). Proc. Am. Math. Soc. 97(3), 465–473 (1986)
Ball, K.: Logarithmically concave functions and sections of convex sets in \(\mathbb R^n\). Studia Math. 88(1), 69–84 (1988)
Ball, K.: Some remarks on the geometry of convex sets. Geometric aspects of functional analysis, Israel seminar (1986/87). Springer Lecture Notes in Math., vol. 1317, pp. 224–231 (1988)
Ball, K.: Normed spaces with a weak Gordon-Lewis property. Lecture Notes in Mathematics, vol. 1470, pp. 36–47. Springer (1991)
Ball, K., Nguyen, V.H.: Entropy jumps for isotropic log-concave random vectors and spectral gap. Studia Math. 213(1), 81–96 (2012)
Barthe, F., Guédon, O., Mendelson, S., Naor, A.: A probabilistic approach to the geometry of the \(l^ n_ p\)-ball. Ann. Probab. 33, 480–513 (2005)
Barthe, F., Fradelizi, M.: The volume product of convex bodies with many hyperplane symmetries. Am. J. Math. 135(2), 311–347 (2013)
Brazitikos, S., Giannopoulos, A., Valettas, P., Vritsiou, B.-H.: Geometry of isotropic convex bodies. American Mathematical Society, Providence, RI (2014)
Berwald, L.: Verallgemeinerung eines Mittelwertsatzes von J. Favard für positive konkave Funktionen. Acta Math. 79, 17–37 (1947)
Bobkov, S.G.: On concentration of distributions of random weighted sums. Ann. Prob. 31(1), 195–215 (2003)
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., vol. 1807, pp. 53–69. Springer (2003)
Bobkov, S.G., Koldobsky, A.: On the central limit property of convex bodies. Geometric Aspects of Functional Analysis. Lecture Notes in Math., vol. 1807, pp. 44–52. Springer (2003)
Borell, C.: Inverse Hölder inequalities in one and several dimensions. J. Math. Anal. Appl. 41, 300–312 (1973)
Borell, C.: Complements of Lyapunov’s inequality. Math. Ann. 205, 323–331 (1973)
Borell, C.: Convex measures on locally convex spaces. Ark. Mat. 12, 239–252 (1974)
Borell, C.: The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30(2), 207–216 (1975)
Bourgain, J.: On high-dimensional maximal functions associated to convex bodies. Am. J. Math. 108(6), 1467–1476 (1986)
Bourgain, J.: Geometry of Banach spaces and harmonic analysis. In: Proceedings of the International Congress of Mathematicians (ICM Berkeley 1986), pp. 871–878. Amer. Math. Soc. (1987)
Bourgain, J.: On the distribution of polynomials on high-dimensional convex sets. Geometric Aspects of Functional Analysis. Israel Seminar (1989–90). Lecture Notes in Math., vol. 1469, pp. 127–137. Springer (1991)
Bourgain, J.: On the isotropy-constant problem for “Psi-2” bodies. Geom. Aspects of Funct. Anal. Israel Seminar (2001–02). Lecture Notes in Math., vol. 1807, pp. 114–121. Springer (2002)
Bourgain, J., Milman, V.D.: New volume ratio properties for convex symmetric bodies in \(\mathbb R^n\). Invent. Math. 88(2), 319–340 (1987)
Bourgain, J., Klartag, B., Milman V.: A reduction of the slicing problem to finite volume ratio bodies. C. R. Acad. Sci. Paris Ser. I 336, 331–334 (2003)
Bourgain, J., Klartag, B., Milman V.: Symmetrization and isotropic constants of convex bodies. Geom. aspects of Funct. Anal. Israel Seminar. Springer Lecture Notes in Math., vol. 1850, pp. 101–116. Springer (2004)
Brehm, U., Voigt, J.: Asymptotics of cross sections for convex bodies. Beiträge Algebra Geom. 41(2), 437–454 (2000)
Busemann, H.: A theorem on convex bodies of the Brunn-Minkowski type. Proc. Natl. Acad. Sci. USA 35, 27–31 (1949)
Busemann, H., Petty, C.M.: Problems on convex bodies. Mathematica Scandinavica 4(1), 88–94 (1956)
Diaconis, P., Freedman, D.: Asymptotics of graphical projection pursuit. Ann. Stat. 12(3), 793–815 (1984)
Eldan, R.: Thin shell implies spectral gap up to polylog via a stochastic localization scheme. Geom. Funct. Anal. (GAFA) 23(2), 532–569 (2013)
Eldan, R., Klartag, B.: Pointwise estimates for marginals of convex bodies. J. Funct. Anal. 254(8), 2275–2293 (2008)
Eldan, R., Klartag, B.: Approximately Gaussian marginals and the hyperplane conjecture. Concentration, Functional Inequalities and Isoperimetry. Contemp. Math., vol. 545, pp. 55–68. Amer. Math. Soc. (2011)
Fernique, X: Régularité de processus gaussiens. Invent. Math. 12, 304–320 (1971)
Figiel, T., Tomczak-Jaegermann, N.: Projections onto Hilbertian subspaces of Banach spaces. Isr. J. Math. 53, 155–171 (1979)
Fleury, B.: Concentration in a thin euclidean shell for log-concave measures. J. Funct. Anal. 259, 832–841 (2010)
Fradelizi, M.: Hyperplane sections of convex bodies in isotropic position. Beiträge Algebra Geom. 40(1), 163–183 (1999)
Gardner, R.J.: Geometric tomography. Second edition. Encyclopedia of Mathematics and its Applications, vol. 58. Cambridge University Press, New York (2006)
Giannopoulos, A., Pajor, A., Paouris, G.: A note on subgaussian estimates for linear functionals on convex bodies. Proc. Am. Math. Soc. 135(8), 2599–2606 (2007)
Giannopoulos, A., Paouris, G., Valettas, P.: On the existence of subgaussian directions for log-concave measures. Concentration, Functional Inequalities and Isoperimetry. Contemp. Math., vol. 545, pp. 103–122. Amer. Math. Soc. (2011)
Giannopoulos, A., Paouris, G., Vritsiou, B.-H.: The isotropic position and the reverse Santaló inequality. Isr. J. Math. 203(1), 1–22 (2014)
Gromov, M., Milman, V.D.: Brunn theorem and a concentration of volume phenomena for symmetric convex bodies. Israel Seminar on Geometrical Aspects of Functional Analysis (1983/84), 12 p. Tel Aviv Univ., Tel Aviv (1984)
Guédon, O., Milman, E.: Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal. (GAFA) 21(5), 1043–1068 (2011)
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)
Junge, M.: Hyperplane conjecture for quotient spaces of L p. Forum Math. 6(5), 617–635 (1994)
Junge, M.: Proportional subspaces of spaces with unconditional basis have good volume properties. Geometric Aspects of Functional Analysis (Israel, 1992–1994). Oper. Theory Adv. Appl., vol. 77, pp. 121–129. Birkhäuser (1995)
Kannan, R., Lovász, L., Simonovits M.: Isoperimetric problems for convex bodies and a localization lemma. J. Discr. Comput. Geom. 13, 541–559 (1995)
Kashin, B.S.: The widths of certain finite-dimensional sets and classes of smooth functions. Izv. Akad. Nauk SSSR Ser. Mat. 41(2), 334–351 (1977). English translation: Math. USSR-Izv. 11(2), 317–333 (1978)
Klartag, B.: An isomorphic version of the slicing problem. J. Funct. Anal. 218(2), 372–394 (2005)
Klartag, B.: On convex perturbations with a bounded isotropic constant. Geom. and Funct. Anal. (GAFA) 16(6), 1274–1290 (2006)
Klartag, B.: Uniform almost sub-gaussian estimates for linear functionals on convex sets. Algebra Anal. (St. Petersburg Math. J.) 19(1), 109–148 (2007)
Klartag, B.: A central limit theorem for convex sets. Invent. Math. 168, 91–131 (2007)
Klartag, B.: Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245, 284–310 (2007)
Klartag, B.: High-dimensional distributions with convexity properties. Proc. of the Fifth Euro. Congress of Math., Amsterdam, July 2008, pp. 401–417. Eur. Math. Soc. Publishing House (2010)
Klartag, B.: Isotropic constants and Mahler volumes. Adv. Math. 330, 74–108 (2018)
Klartag, B., Koldobsky, A.: An example related to the slicing inequality for general measures. J. Funct. Anal. 274(7), 2089–2112 (2018)
Klartag, B., Lehec, J.: Bourgain’s slicing problem and KLS isoperimetry up to polylog. Preprint, ar**v:2203.15551
Klartag, B., Kozma, G.: On the hyperplane conjecture for random convex sets. Isr. J. Math. 170, 253–268 (2009)
Klartag, B., Milman, V.D.: Isomorphic Steiner symmetrization. Invent. Math. 153(3), 463–485 (2003)
Klartag, B., Milman, V.: Rapid Steiner symmetrization of most of a convex body and the slicing problem. Combin. Probab. Comput. 14(5–6), 829–843 (2005)
Klartag, B., Milman, V.: Geometry of log-concave functions and measures. Geom. Dedicata 112, 169–182 (2005)
Klartag, B., Milman, E.: Centroid bodies and the logarithmic Laplace transform – a unified approach. J. Funct. Anal. 262(1), 10–34 (2012)
Koldobsky, A.: Fourier analysis in convex geometry. Mathematical Surveys and Monographs, vol. 116. American Mathematical Society, Providence, RI (2005)
König, H., Meyer, M., Pajor, A.: The isotropy constants of the Schatten classes are bounded. Math. Ann. 312, 773–783 (1998)
Kuperberg, G.: From the Mahler conjecture to Gauss linking integrals. Geom. Funct. Anal. (GAFA) 18(3), 870–892 (2008)
Lee, Y.T., Vempala, S.: Eldan’s stochastic localization and the KLS hyperplane conjecture: an improved lower bound for expansion. Symposium on Foundations of Computer Science (FOCS), pp. 998–1007. IEEE Computer Soc. (2017)
Lee, Y.T., Vempala, S.: The Kannan-Lovász-Simonovits conjecture. Current Developments in Mathematics 2017, pp. 1–36. Int. Press (2019)
Lewis, D.R.: Ellipsoids defined by Banach ideal norms. Mathematica 26, 18–29 (1979)
Litvak, A.E., Milman, V.D., Schechtman, G.: Averages of norms and quasi-norms. Math. Ann. 312(1), 95–124 (1998)
Loomis, L.H., Whitney, H.: An inequality related to the isoperimetric inequality. Bull. Am. Math. Soc. 55, 961–962 (1949)
Lutwak, E., Zhang, G.: Blaschke-Santaló inequalities. J. Differ. Geom. 47(1), 1–16 (1997)
Magazinov, A.: A proof of a conjecture by Haviv, Lyubashevsky and Regev on the second moment of a lattice Voronoi cell. Adv. Geom. 20(1), 117–120 (2020)
Mahler, K.: Ein Übertragungsprinzip für konvexe körper. Časopis Pest Mat. Fys. 68, 93–102 (1939)
Meyer, M.: Convex bodies with minimal volume product in \(\mathbb R^2\). Monatsh. Math. 112(4), 297–301 (1991)
Meyer, M., Pajor, A.: On Santaló’s inequality. Geometric Aspects of Functional Analysis (1987–88). Lecture Notes in Math., vol. 1376, pp. 261–263. Springer (1989)
Meyer, M., Pajor, A.: On the Blaschke-Santaló inequality. Arch. Math. 55(1), 82–93 (1990)
Milman, E.: Dual mixed volumes and the slicing problem. Adv. Math. 207, 566–598 (2006)
Milman, V.D.: Inégalité de Brunn-Minkowski inverse et applications á la théorie locale des espaces normés. [An inverse form of the Brunn-Minkowski inequality, with applications to the local theory of normed spaces] C. R. Acad. Sci. Paris Sér. I Math. 302(1), 25–28 (1986)
Milman, V.D.: Isomorphic symmetrization and geometric inequalities. Geometric Aspects of Functional Analysis (1986/87). Lecture Notes in Math., vol. 1317, pp. 107–131. Springer (1988)
Milman, V.D.: Dvoretzky’s theorem—thirty years later. Geom. Funct. Anal. 2(4), 455–479 (1992)
Milman, V.D., Schechtman, G.: Asymptotic theory of finite-dimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Mathematics, vol. 1200. Springer (1986)
Milman, V.D., Pajor, A.: 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, pp. 64–104. Springer (1989)
Milman, V.D., Pajor, A.: Entropy and asymptotic geometry of non-symmetric convex bodies. Adv. Math. 152(2), 314–335 (2000)
Nazarov, F.: The Hörmander proof of the Bourgain-Milman theorem. Geometric Aspects of Functional Analysis. Lecture Notes in Math., vol. 2050, pp. 335–343. Springer (2012)
Rogers, C.A., Shephard, G.C.: The difference body of a convex body. Arch. Math. 8, 220–233 (1957)
Paouris, G.: ψ 2-estimates for linear functionals on zonoids. Geometric Aspects of Functional Analysis. Lecture Notes in Math., vol. 1807, 211–222. Springer, Berlin (2003)
Paouris, G.: On the Ψ2 behavior of linear functionals on isotropic convex bodies. Studia Math. 168(3), 285–299 (2005)
Paouris, G.: Concentration of mass on convex bodies. Geom. Funct. Anal. (GAFA) 16(5), 1021–1049 (2006)
Pisier, G.: Holomorphic semigroups and the geometry of Banach spaces. Ann. Math. 115(2), 375–392 (1982)
Pisier, G.: The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics, Vol. 94. Cambridge University Press, Cambridge (1989)
Rademacher, L.: A simplicial polytope that maximizes the isotropic constant must be a simplex. Mathematika 62(1), 307–320 (2016)
Sudakov, V.N.: Typical distributions of linear functionals in finite-dimensional spaces of high-dimension. (Russian) Dokl. Akad. Nauk. SSSR 243(6), 1402–1405 (1978). English translation in Soviet Math. Dokl. 19, 1578–1582 (1978)
Szarek, S.J.: On Kashin’s almost Euclidean orthogonal decomposition of \(\ell ^1_n\). Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26(8), 691–694 (1978)
Szarek, S., Tomczak-Jaegermann, N.: On nearly Euclidean decomposition for some classes of Banach spaces. Compositio Math. 40(3), 367–385 (1980)
Talagrand, M.: The generic chaining. Upper and Lower Bounds of Stochastic Processes. Springer (2005)
Talagrand, M.: Upper and lower bounds for stochastic processes. Modern Methods and Classical Problems. Springer (2014)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Klartag, B., Milman, V. (2022). The Slicing Problem by Bourgain. In: Avila, A., Rassias, M.T., Sinai, Y. (eds) Analysis at Large. Springer, Cham. https://doi.org/10.1007/978-3-031-05331-3_9
Download citation
DOI: https://doi.org/10.1007/978-3-031-05331-3_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-05330-6
Online ISBN: 978-3-031-05331-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)