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\(L_p\) Busemann–Petty centroid inequality in hyperbolic and spherical spaces

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The \(L_p\) Busemann–Petty centroid inequality states that if K is a compact domain in Euclidean space \(\mathbb {R}^n\), then, for \(p\ge 1\),

$$\begin{aligned} \textrm{vol}(\Gamma _pK)\ge \textrm{vol}(K), \end{aligned}$$

with equality if and only if K is an ellipsoid centered at the origin. We extend it to hyperbolic and spherical spaces, as well as general measure spaces.

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References

  1. Berenstein, C.A., Casadio Tarabusi, E.: Integral geometry in hyperbolic spaces and electrical impedance tomography. SIAM J. Appl. Math. 56(3), 755–764 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blaschke, W.: Affine Geometrie IX: Verschiedene Bemerkungen und Aufgaben. Ber. Verh. Sachs. Akad. Leipz. Math. Phys. Kl. 69, 412–420 (1917)

    MATH  Google Scholar 

  3. Brazitikos, S., Giannopoulos, A., Valettas, P., Vritsiou, B.-H.: Geometry of Isotropic Convex Bodies. Mathematical Surveys and Monographs, vol. 196. American Mathematical Society, Providence (2014)

  4. Campi, S., Gronchi, P.: The \(L^p\)-Busemann–Petty centroid inequality. Adv. Math. 167(1), 128–141 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dann, S., Kim, J., Yaskin, V.: Busemann’s intersection inequality in hyperbolic and spherical spaces. Adv. Math. 326, 521–560 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry—Methods and Applications. Part I. The Geometry of Surfaces, Transformation groups, and Fields. Second edition. Translated from the Russian by Robert G. Burns. Graduate Texts in Mathematics, 93. Springer, New York (1992)

  7. Fillastre, F.: Fuchsian convex bodies: basics of Brunn–Minkowski theory. Geom. Funct. Anal. 23(1), 295–333 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fillastre, F., Izmestiev, I.: Shapes of polyhedra, mixed volumes and hyperbolic geometry. Mathematika 63(1), 124–183 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Firey, W.J.: \(p\)-means of convex bodies. Math. Scand. 10, 17–24 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gao, F., Hug, D., Schneider, R.: Intrinsic volumes and polar sets in spherical space. Math. Notae 41(2001/02), 159–176 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Gardner, R.J.: Geometric Tomography, 2nd edn. Cambridge University Press, New York (2006)

    Book  MATH  Google Scholar 

  12. Haberl, C., Schuster, F.E.: General \(L_p\) affine isoperimetric inequalities. J. Differ. Geom. 83(1), 1–26 (2009)

    Article  MATH  Google Scholar 

  13. Haddad, J.: A convex body associated to the Busemann random simplex inequality and the Petty conjecture. J. Funct. Anal. 281(7), Paper No. 109118, 28 pp. (2021)

  14. Haddad, J., Jiménez, C.H., Montenegro, M.: Sharp affine Sobolev type inequalities via the \(L_p\) Busemann–Petty centroid inequality. J. Funct. Anal. 271(2), 454–473 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Haddad, J.E., Jiménez, C.H., da Silva, L.A.: An \(L_p\)-functional Busemann–Petty centroid inequality. Int. Math. Res. Not. IMRN 10, 7947–7965 (2021)

    Article  MATH  Google Scholar 

  16. Helgason, S.: Groups and Geometric Analysis. Pure and Applied Mathematics, vol. 113. Academic Press Inc, Orlando (1984)

  17. Lieb, E.H., Loss, M.: Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence (2001)

  18. Lutwak, E.: On some affine isoperimetric inequalities. J. Differ. Geom. 23(1), 1–13 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lutwak, E.: Selected affine isoperimetric inequalities. In: Handbook of Convex Geometry, Vol. A, B, pp. 151–176. North-Holland, Amsterdam

  20. Lutwak, E.: The Brunn–Minkowski–Firey theory. I. Mixed volumes and the Minkowski problem. J. Differ. Geom. 38(1), 131–150 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lutwak, E.: The Brunn–Minkowski–Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118(2), 244–294 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lutwak, E., Yang, D., Zhang, G.: \(L_p\) affine isoperimetric inequalities. J. Differ. Geom. 56(1), 111–132 (2000)

    Article  MATH  Google Scholar 

  23. Petty, C.M.: Centroid surfaces. Pac. J. Math. 11, 1535–1547 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  24. Petty, C.M.: Isoperimetric problems. In: Proceedings of the Conference on Convexity and Combinatorial Geometry (Univ. Oklahoma, Norman, Okla.), pp. 26–41. Dept. Math., Univ. Oklahoma, Norman, OK (1971)

  25. Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol. 149. Springer, New York (1994)

  26. Xia, Y.: The dual Brunn–Minkowski inequalities in spherical and hyperbolic spaces. Bull. Iran. Math. Soc. 48(5), 2843–2854 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yaskin, V.: The Busemann–Petty problem in hyperbolic and spherical spaces. Adv. Math. 203(2), 537–553 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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  1. School of Mathematics and Statistics, Changshu Institute of Technology, Suzhou, 215500, China

    Songjun Lv, Shengjie Zhang & Shiyu Shen

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  1. Songjun Lv
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Lv, S., Zhang, S. & Shen, S. \(L_p\) Busemann–Petty centroid inequality in hyperbolic and spherical spaces. Arch. Math. 120, 651–663 (2023). https://doi.org/10.1007/s00013-023-01856-z

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