The Busemann-Petty Problem

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 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.