## 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: Here,

$$s\left( K \right) = \frac{1}{{{\omega _{n - 1}}}}\int_{\partial \left( B \right)} {\bar v\left( {{P_u}\left( K \right)} \right)d\lambda \left( u \right)} .$$

*s*(*K*) denotes the surface area of a convex body*K*⊂*R*^{ n }, \(\bar v\left( X \right)\) denotes the (*n*− 1)-dimensional “area” of a set*X*⊂*R*^{ n }^{−1},*P*_{ u }denotes the orthogonal projection from*R*^{ n }to the hyperplane*H*_{ u }= {*x*∈*R*^{ 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*) =*K*⋂*H*_{ u }*)*proved to be unexpectedly and extremely difficult.## Preview

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## Copyright information

© Springer-Verlag New York, Inc. 1996