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The role of the Rogers–Shephard inequality in the characterization of the difference body

  • Judit Abardia-Evéquoz and Eugenia Saorín Gómez EMAIL logo
Published/Copyright: December 2, 2016
Forum Mathematicum
From the journal Forum Mathematicum Volume 29 Issue 6

Abstract

The Rogers–Shephard and Brunn–Minkowski inequalities provide upper and lower bounds for the volume of the difference body in terms of the volume of the body itself. In this work it is shown that the difference body operator is the only continuous and GL⁡(n)-covariant operator from the space of convex bodies to the origin-symmetric ones which satisfies a Rogers–Shephard-type inequality. This is a consequence of a more general classification result for operators on convex bodies of the type a⁢K+b⁢(-K).

Keywords: Difference body; Rogers–Shephard inequality; Minkowski valuation
MSC 2010: 52A20; 52B45; 52A40

Communicated by Karl Strambach

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: AB 584/1-1

Award Identifier / Grant number: AB 584/1-2

Funding source: Ministerio de Economía y Competitividad

Award Identifier / Grant number: MTM2012-34037

Funding source: Fundación Séneca

Award Identifier / Grant number: 19901/GERM/15

Funding statement: The first author is supported by grants AB 584/1-1 and AB 584/1-2 of the Deutsche Forschungsgemeinschaft (DFG). The second author is partially supported by the project MTM2012-34037 of the Ministerio de Economía y Competitividad (MINECO/FEDER), and by the project 19901/GERM/15 of the Fundación Séneca, http://dx.doi.org/10.13039/100007801.

Acknowledgements

The authors would like to thank the referees for their careful review of the manuscript. We do highly appreciate the constructive comments and the suggestions which have undoubtedly helped to improve this work.

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Received: 2016-4-26
Revised: 2016-10-10
Published Online: 2016-12-2
Published in Print: 2017-11-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. The role of the Rogers–Shephard inequality in the characterization of the difference body
  3. Test vectors for local periods
  4. The local Langlands conjecture for p-adic GSpin4, GSpin6, and their inner forms
  5. A family of irretractable square-free solutions of the Yang–Baxter equation
  6. On Sylow permutable subgroups of finite groups
  7. Using Steinberg algebras to study decomposability of Leavitt path algebras
  8. The ℓ∞-semi-norm on uniformly finite homology
  9. Minimal potential results for Schrödinger equations with Neumann boundary conditions
  10. On a unique continuation for Aharonov–Bohm magnetic Schrödinger equation
  11. Sup-norm bounds for Eisenstein series
  12. Permutohedral structures on E2-operads
  13. On the complete integrability of the periodic quantum Toda lattice
  14. Fans, decision problems and generators of free abelian ℓ-groups
  15. Standard cocycles: Variations on themes of C. Kassel’s and R. Wilson’s
  16. Crossed modules as maps between connected components of topological groups
  17. A local converse theorem for U(2,2)
  18. Probabilistic trace and Poisson summation formulae on locally compact abelian groups
https://doi.org/10.1515/forum-2016-0101
Keywords for this article
Difference body; Rogers–Shephard inequality; Minkowski valuation

Articles in the same Issue

  1. Frontmatter
  2. The role of the Rogers–Shephard inequality in the characterization of the difference body
  3. Test vectors for local periods
  4. The local Langlands conjecture for p-adic GSpin4, GSpin6, and their inner forms
  5. A family of irretractable square-free solutions of the Yang–Baxter equation
  6. On Sylow permutable subgroups of finite groups
  7. Using Steinberg algebras to study decomposability of Leavitt path algebras
  8. The ℓ∞-semi-norm on uniformly finite homology
  9. Minimal potential results for Schrödinger equations with Neumann boundary conditions
  10. On a unique continuation for Aharonov–Bohm magnetic Schrödinger equation
  11. Sup-norm bounds for Eisenstein series
  12. Permutohedral structures on E2-operads
  13. On the complete integrability of the periodic quantum Toda lattice
  14. Fans, decision problems and generators of free abelian ℓ-groups
  15. Standard cocycles: Variations on themes of C. Kassel’s and R. Wilson’s
  16. Crossed modules as maps between connected components of topological groups
  17. A local converse theorem for U(2,2)
  18. Probabilistic trace and Poisson summation formulae on locally compact abelian groups
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