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Monatshefte für Mathematik

, Volume 112, Issue 1, pp 43–60 | Cite as

Stability properties of Cauchy's surface area formula

  • H. Groemer
Article

Abstract

Cauchy's surface area formula expresses the surface area of ad-dimensional convex body in terms of the mean value of the volume of its orthogonal projections onto (d−1)-dimensional linear subspaces. We consider here averages of the same kind as those in Cauchy's formula but with respect to some direction dependent density function and investigate the stability problem whether the density must be close to 1 if the formula produces approximately the correct surface area. It will be shown that this relationship between surface area and density is, in general, unstable; but if the density function satisfies suitable regularity conditions, then explicit stability estimates can be obtained.

Keywords

Density Function Orthogonal Projection Convex Body Regularity Condition Stability Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper. Berlin: Springer. 1934.Google Scholar
  2. [2]
    Bourgain, J., Lindenstrauss, J.: Projection bodies. In: Geometric Aspects of Functional Analysis. Lect. Notes Math.1317. (J. Lindenstrauss and V. D. Milman, Eds.) Berlin-Heidelberg-New York: Springer. pp. 250–270. 1988.Google Scholar
  3. [3]
    Campi, S.: On the reconstruction of a function on a sphere by its integrals over great circles. Boll. Un. Mat. Ital. C(5)18, 195–215 (1981).Google Scholar
  4. [4]
    Campi, S.: On the reconstruction of a star-shaped body from its “half-volumes”. J. Austral. Math. Soc. (Ser. A)37, 243–257 (1984).Google Scholar
  5. [5]
    Campi, S.: Reconstructing a convex surface from certain measurements of its projections. Boll. Un. Mat. Ital. (6)5-B, 945–959 (1986).Google Scholar
  6. [6]
    Campi, S.: Recovering a centered convex body from the areas of its shadows: a stability estimate. Ann. Mat. Pura Appl.151, 289–302 (1988).Google Scholar
  7. [7]
    Goodey, P. R., Groemer, H.: Stability results for first order projection bodies. Proc. Amer. Math. Soc.109, 1103–1114 (1990).Google Scholar
  8. [8]
    Groemer, H.: Stability theorems for projections and central symmetrization. Arch. d. Math. To appear.Google Scholar
  9. [9]
    Hochstadt, H.: The Functions of Mathematical Physics. New York: Dover. 1986.Google Scholar
  10. [10]
    Müller, C.: Spherical Harmonics. Lect. Notes Math. 17, Berlin-Heidelberg-New York: Springer. 1966.Google Scholar
  11. [11]
    Petty, C. M.: Centroid surfaces. Pacific J. Math.11, 1535–1547 (1961).Google Scholar
  12. [12]
    Santaló, L. A.: Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley. 1976.Google Scholar
  13. [13]
    Schneider, R.: Zu einem Problem von Shephard über die Projektionen konvexer Körper. Math. Zeitschr.101, 71–82 (1967).Google Scholar
  14. [14]
    Schneider, R.: Boundary structure and curvature of convex bodies. In: Contributions to Geometry. Proc. of Geom. Symp., Siegen 1978. Basel-Boston-Stuttgart: Birkhäuser. 1979.Google Scholar
  15. [15]
    Schneider, R.: Stability in the Aleksandrov-Fenchel-Jessen Theorem. Mathematika36, 50–59 (1989).Google Scholar
  16. [16]
    Seeley, R. T.: Spherical harmonics. Am. Math. Monthly73, 115–121 (1966).Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • H. Groemer
    • 1
  1. 1.Department of MathematicsThe University of ArizonaTucsonUSA

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