Shephard's problem

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?Template:Sfn

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 Template: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

$V_k({\pi }_k(K))\le V_k({\pi }_k(L))\text{ for all }1\le k<n⟹V_n(K)\le V_n(L).$

V_k(Template:Pi_k(K)) is sometimes known as the brightness of K and the function V_k o Template: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.Template:SfnTemplate:Sfn The solution of Shephard's problem requires Minkowski's first inequality for convex bodies and the notion of projection bodies of convex bodies.

See also

• Busemann–Petty problem