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On the asymptotics of the principal moments of inertia of a convex body in the isotropic state


We recall Bourgain’s conjecture on the upper bound for the main moments of inertia of a multidimensional body in the isotropic state. We also recall the currently known bounds obtained by Bourgain and Klartag. Then we evaluate the moments of inertia of the multidimensional ball and of the multidimensional cube, and state a conclusive comment corroborating the Bourgain’s conjecture.

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  1. 1.

    K. Ball. “An elementary introduction to modern convex geometry. Flavors of geometry,” Math. Sci. Res. Inst. Publ., 31, 1–58 (1997).

    MATH  Google Scholar 

  2. 2.

    J. Bourgain. On the distribution of polynomials on high dimensional convex sets. Geometric Aspects of Functional Analysis (1989–90), Lecture Notes in Math., 1469, Springer, Berlin, pp. 127–137, 1991.

  3. 3.

    J. Bourgain, B. Klartag, and V. Milman. Symmetrization and isotropic constants of convex bodies. Geometric aspects of functional analysis, Lecture Notes in Math., 1850, Springer, Berlin, pp. 101–115, 2004.

  4. 4.

    B. Klartag. “On convex perturbations with a bounded isotropic constant,” Geom. funct. anal., 16(6), 1274–1290 (2006). See also Klartag’s web-page for related more recent results.

  5. 5.

    L. D. Landau and E. M. Lifshitz. Short course in theoretical physics. Book 1. Mechanics. Electrodynamics [in Russian]. Moscow, Nauka, 1969.

  6. 6.

    V. D. Milman. Geometrization of Probability. Progress in Mathematics, 265, 2008.

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Correspondence to Vladimir A. Zorich.

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On the 80th anniversary of Vladimir Yakovlevich Gutlyanskii

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 18, No. 3, pp. 428–432, July–September, 2021.

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Zorich, V.A. On the asymptotics of the principal moments of inertia of a convex body in the isotropic state. J Math Sci 259, 104–107 (2021).

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  • Convex body
  • tensor of inertia
  • isotropic state
  • isotropy constant
  • Bourgain’s conjecture
  • slicing problem
  • hyperplane conjecture