A Reverse Rogers–Shephard Inequality for Log-Concave Functions

Article
  • 8 Downloads

Abstract

We will prove a reverse Rogers–Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume estimates for polars of \(\ell _p\)-diferences of convex bodies under the condition that their polar bodies have opposite barycenters.

Keywords

Rogers–Shephard inequality Log-concave functions Log-concave measures Geometric inequalities Functional inequalities 

Mathematics Subject Classification

52A20 39B62 

Notes

Acknowledgements

Partially suppored by MINECO Project MTM2016-77710-P.

References

  1. 1.
    Alonso-Gutiérrez, D., González Merino, B., Jiménez, C.H., Villa, R.: Rogers-Shephard inequality for log-concave functions. J. Func. Anal. 271(11), 3269–3299 (2016)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alonso-Gutiérrez, D., Jiménez, C.H., Villa, R.: Brunn-Minkowski and Zhang inequalities for convolution bodies. Adv. Math. 238, 50–69 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Artstein-Avidan, S., Einhorn, K., Florentin, D.I., Ostrover, Y.: On Godbersen’s conjecture. Geom. Dedicata 178(1), 337–350 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Artstein-Avidan, S., Giannopoulos, A., Milman, V.D.: Asymptotic Geometric Analysis, Part 1. Mathematical Surveys and Monographs, vol. 122. American Mathematical Society, Providence, RI (2015)CrossRefMATHGoogle Scholar
  5. 5.
    Ball, K.: Logarithmically concave functions and sections of convex sets in \({\mathbb{R}}^n\). Studia Math. 88(1), 69–84 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brazitikos, S., Giannopoulos, A., Valettas, P., Vritsiou B.-H.:, Mathematical Surveys and Monographs, vol. 196. American Mathematical Society, Providence, RI, (2014). ISBN 978-1-4704-1456-6Google Scholar
  7. 7.
    Firey, W.J.: Polar means of convex bodies and a dual to the Brunn-Minkowski theorem. Can. Math. J. 13, 444–453 (1961)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Firey, W.J.: Mean cross-section measures of harmonic means of convex bodies. Pac. J. Math. 11, 1263–1266 (1961)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hernández Cifre, M.A., Yepes, Nicolás J.: On Brunn-Minkowski type inequalities for polar bodies. J. Geom. Anal. 26(1), 143–155 (2016)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Milman, V.D., Pajor, A.: Entropy and asymptotic geometry of non-symmetric convex bodies. Adv. Math. 152(2), 314–335 (2000)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Rogers, C.A., Shephard, G.C.: The difference body of a convex body. Arch. Math. 8, 220–233 (1957)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Rogers, C.A., Shephard, G.C.: Convex bodies associated with a given convex body. J. Lond. Math. Soc. 33, 270–281 (1958)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Área de Análisis Matemático, Departamento de Matemáticas, Facultad de Ciencias, IUMAUniversidad de ZaragozaZaragozaSpain

Personalised recommendations