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(K) denotes the surface area of a convex body K ⊂ Rn, \(\bar v\left( X \right)\) denotes the (n − 1)-dimensional “area” of a set X ⊂ Rn−1, P u denotes the orthogonal projection from Rn to the hyperplane H u = {x ∈ Rn: 〈x, u〉 = 0} determined by a unit vector u of Rn, 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.
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© 1996 Springer-Verlag New York, Inc.
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Zong, C., Dudziak, J.J. (1996). The Busemann-Petty Problem. In: Dudziak, J.J. (eds) Strange Phenomena in Convex and Discrete Geometry. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8481-6_6
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DOI: https://doi.org/10.1007/978-1-4613-8481-6_6
Publisher Name: Springer, New York, NY
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