Skip to main content
Log in

A Remarkable Measure Preserving Diffeomorphism between two Convex Bodies in ℝn

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript


We prove that for any two convex open bounded bodies K and T there exists a diffeomorphism f : K → T preserving volume ratio (i.e. with constant determinant of the Jacobian) and such that the Minkowski sum K + T { x + f (x) | x ∈ K }. As an application of this method, we prove some of the Alexandov–Fenchel inequalities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Alexandrov, A. D.: Zur Theorie gemischter Volumina von konvexen Körpern IV, Die gemischten Diskriminanten und die gemischten Volumina, Mat. Sb. SSSR 3 (1938), 227-251.

    Google Scholar 

  2. Anderson, R. D. and Klee, V. L.: Convex functions and upper semicontinuous collections. Duke Math. J. 19 (1952), 349-357.

    Google Scholar 

  3. Bourgain, J.: On the distribution of polynomials on high dimensional convex sets, in: GAFA Seminar Notes '89–'90, Lecture Notes in Math. 1469, Springer-Verlag, New York, 1991, pp. 127-137.

    Google Scholar 

  4. Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375-417.

    Google Scholar 

  5. Burago, Yu. D. and Zalgaller, V. A.: Geometric Inequalities, Springer Ser. Soviet Math., Springer-Verlag, Berlin, 1988.

    Google Scholar 

  6. Busemann, H.: Convex Surfaces, Interscience, New York, 1958.

    Google Scholar 

  7. Caffarelli, L. A.: A localization property of viscosity solutions to the Monge-Ampère equation and their strict convexity, Ann. Math. 131 (1990), 129-134.

    Google Scholar 

  8. Caffarelli, L. A.: A-priori Estimates and the Geometry of the Monge-Ampère Equation, Park City/IAS Math. Ser. II, Amer. Math. Soc., Providence, 1992.

    Google Scholar 

  9. Caffarelli, L. A.: Some regularity properties of solutions of Monge-Ampère equation, Comm. Pure Appl. Math. 44 (1991), 965-969.

    Google Scholar 

  10. Caffarelli, L. A.: Private communication.

  11. Gromov, M.: Convex sets and Kähler manifolds, in Advances in Differential Geometry and Topology, World Scientific, Teaneck, NJ, 1990, pp. 1-38.

    Google Scholar 

  12. Hörmander, L.: Notions of Convexity, Progress in Math. 127, Birkhäuser, Boston, 1994.

    Google Scholar 

  13. McCann, R. J.: Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80(2) (1995), 309-323.

    Google Scholar 

  14. McCann, R. J.: A convexity principle for interacting gases, Adv. in Math. 128(1) (1997), 153-179.

    Google Scholar 

  15. Milman, V. D. and Schechtman, G.: Asymptotic theory of finite dimensional normed spaces (with appendix by M. Gromov), Lecture Notes in Math. 1200, Springer-Verlag, New York, 1986.

    Google Scholar 

  16. Rockafellar, R. T.: Convex Analysis, Princeton University Press, Princeton, 1972.

    Google Scholar 

  17. Schneider, R.: Convex bodies: the Brunn-Minkowski theory, Encyclopedia Math. Appl. 44, Cambridge University Press, Cambridge, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and permissions

About this article

Cite this article

Alesker, S., Dar, S. & Milman, V. A Remarkable Measure Preserving Diffeomorphism between two Convex Bodies in ℝn . Geometriae Dedicata 74, 201–212 (1999).

Download citation

  • Issue Date:

  • DOI: