Shephard's problem

{{No footnotes|date=November 2019}}

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?{{sfn|Shephard|1964}}

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 {{pi}}_k : Rⁿ → Π_k is a projection of Rⁿ onto some k-dimensional hyperplane Π_k (not necessarily a coordinate hyperplane) and V_k denotes k-dimensional volume, Shephard's problem is to determine the truth or falsity of the implication

:

V_k({{pi}}_k(K)) is sometimes known as the brightness of K and the function V_k o {{pi}}_k 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 ≥ 3.{{sfn|Petty|1967}}{{sfn|Schneider|1967}} The solution of Shephard's problem requires Minkowski's first inequality for convex bodies and the notion of projection bodies of convex bodies.