In mathematics, Shephard's problem, is the following geometrical question asked by Geoffrey Colin Shephard in 1964: if K and L are centrally symmetric convex bodies in n-dimensional Euclidean space such that whenever K and L are projected onto a hyperplane, the volume of the projection of K is smaller than the volume of the projection of L, then does it follow that the volume of K is smaller than that of L?
In this case, "centrally symmetric" means that the reflection of K in the origin, −K, is a translate of K, and similarly for L. If <sub>k</sub> : R<sup>n</sup> â à<sub>k</sub> is a projection of R<sup>n</sup> onto some k-dimensional hyperplane à<sub>k</sub> (not necessarily a coordinate hyperplane) and V<sub>k</sub> denotes k-dimensional volume, Shephard's problem is to determine the truth or falsity of the implication
V<sub>k</sub>(<sub>k</sub>(K)) is sometimes known as the brightness of K and the function V<sub>k</sub> <small>o</small> <sub>k</sub> as a (k-dimensional) brightness function.
In dimensions n = 1 and 2, the answer to Shephard's problem is "yes". In 1967, however, Petty and Schneider showed that the answer is "no" for every n âÂÂ¥ 3. The solution of Shephard's problem requires Minkowski's first inequality for convex bodies and the notion of projection bodies of convex bodies.