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The Busemann-Petty Problem

  • Chuanming Zong
  • James J. Dudziak
Part of the Universitext book series (UTX)

Abstract

The search for relationships between a convex body and its projections or sections has a long history. In 1841, A. Cauchy found that the surface area of a convex body can be expressed in terms of the areas of its projections as follows:
$$s\left( K \right) = \frac{1}{{{\omega _{n - 1}}}}\int_{\partial \left( B \right)} {\bar v\left( {{P_u}\left( K \right)} \right)d\lambda \left( u \right)} .$$
Here, s(K) denotes the surface area of a convex body KR n , \(\bar v\left( X \right)\) denotes the (n − 1)-dimensional “area” of a set XR n −1, P u denotes the orthogonal projection from R n to the hyperplane H u = {xR n : 〈x, u〉 = 0} determined by a unit vector u of R n , and λ denotes surface-area measure on (B). In contrast, the closely related problem of finding an expression for the volume of K in terms of the areas of its projections P u (K) (or the areas of its sections I u (K) = KH u ) proved to be unexpectedly and extremely difficult.

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Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Chuanming Zong
    • 1
  • James J. Dudziak
    • 2
  1. 1.Institute of MathematicsThe Chinese Academy of SciencesBeijingPR China
  2. 2.Department of MathematicsBucknell UniversityLewisburgUSA

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