Illinois Journal of Mathematics

Estimates for the affine and dual affine quermassintegrals of convex bodies

Nikos Dafnis and Grigoris Paouris

Full-text: Open access

Abstract

We provide estimates for suitable normalizations of the affine and dual affine quermassintegrals of a convex body $K$ in $\mathbb{R}^{n}$. These follow by a more general study of normalized $p$-means of projection and section functions of $K$.

Article information

Source
Illinois J. Math., Volume 56, Number 4 (2012), 1005-1021.

Dates
First available in Project Euclid: 6 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1399395821

Digital Object Identifier
doi:10.1215/ijm/1399395821

Mathematical Reviews number (MathSciNet)
MR3231472

Zentralblatt MATH identifier
1335.52008

Subjects
Primary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx] 46B06: Asymptotic theory of Banach spaces [See also 52A23]

Citation

Dafnis, Nikos; Paouris, Grigoris. Estimates for the affine and dual affine quermassintegrals of convex bodies. Illinois J. Math. 56 (2012), no. 4, 1005--1021. doi:10.1215/ijm/1399395821. https://projecteuclid.org/euclid.ijm/1399395821


Export citation

References

  • K. M. Ball, Logarithmically concave functions and sections of convex sets in ${\mathbb R}^n$, Studia Math. 88 (1988), 69–84.
  • J. Bourgain, On the distribution of polynomials on high dimensional convex sets, Geometric aspects of functional analysis (J. Lindenstrauss and V. D. Milman, eds.), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 127–137.
  • J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in ${\mathbb R}^n$, Invent. Math. 88 (1987), no. 2, 319–340.
  • N. Dafnis and G. Paouris, Small ball probability estimates, $\psi_2$-behavior and the hyperplane conjecture, J. Funct. Anal. 258 (2010), 1933–1964.
  • H. Furstenberg and I. Tzkoni, Spherical functions and integral geometry, Israel J. Math. 10 (1971), 327–338.
  • R. J. Gardner, Geometric tomography, Encyclopedia of Mathematics and Its Applications, vol. 58, Cambridge University Press, Cambridge, 1995.
  • A. Giannopoulos, Notes on isotropic convex bodies, Warsaw University Notes, 2003.
  • E. L. Grinberg, Isoperimetric inequalities and identities for $k$-dimensional cross-sections of a convex bodies, Math. Ann. 291 (1991), no. 1, 75–86.
  • B. Klartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. 16 (2006), 1274–1290.
  • B. Klartag and E. Milman, Centroid bodies and the logarithmic Laplace transform–-A unified approach, available at \arxivurlarXiv:1103.2985v1.
  • E. Lutwak, A general isepiphanic inequality, Proc. Amer. Math. Soc. 90 (1984), 415–421.
  • E. Lutwak, Inequalities for Hadwiger's harmonic Quermassintegrals, Math. Ann. 280 (1988), 165–175.
  • E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232–261.
  • E. Lutwak and G. Zhang, Blaschke–Santaló inequalities, J. Differential Geom. 47 (1997), 1–16.
  • R. E. Miles, A simple derivation of a formula of Furstenberg and Tzkoni, Israel J. Math. 14 (1973), 278–280.
  • V. D. Milman, Inegalité de Brunn–Minkowski inverse et applications à la théorie locale des espaces normés, C. R. Acad. Sci. Paris 302 (1986), 25–28.
  • V. D. Milman, Isomorphic symmetrization and geometric inequalities, Geometric aspects of functional analysis (J. Lindenstrauss and V. D. Milman, eds.), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 107–131.
  • V. D. Milman and A. Pajor, Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space, GAFA Seminar 87–89, Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104.
  • V. D. Milman and A. Pajor, Entropy and asymptotic geometry of non-symmetric convex bodies, Adv. Math. 152 (2000), 314–335.
  • V. D. Milman and G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Lecture Notes in Math., vol. 1200, Springer, Berlin, 1986.
  • G. Paouris, On the isotropic constant of non-symmetric convex bodies, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1745, Springer, Berlin, 2000, pp. 239–243.
  • G. Paouris, Concentration of mass on convex bodies, Geom. Funct. Anal. 16 (2006), 1021–1049.
  • G. Paouris, Small ball probability estimates for log-concave measures, Trans. Amer. Math. Soc. 364 (2012), 287–308.
  • G. Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989.
  • C. A. Rogers and G. C. Shephard, The difference body of a convex body, Arch. Math. 8 (1957), 220–233.
  • C. A. Rogers and G. C. Shephard, Convex bodies associated with a given convex body, J. London Math. Soc. 33 (1958), 270–281.
  • R. Schneider, Convex bodies: The Brunn–Minkowski theory, Encyclopedia of Mathematics and Its Applications, vol. 44, Cambridge University Press, Cambridge, 1993.
  • G. Zhang, Restricted chord projection and affine inequalities, Geom. Dedicata 39 (1991), 213–222.