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 πk : Rn → Πk is a projection of Rn onto some k-dimensional hyperplane Πk (not necessarily a coordinate hyperplane) and Vk denotes k-dimensional volume, Shephard's problem is to determine the truth or falsity of the implication

V k ( π k ( K ) ) ≤ V k ( π k ( L ) ) for all 1 ≤ k < n ⟹ V n ( K ) ≤ V n ( L ) . {\displaystyle V_{k}(\pi _{k}(K))\leq V_{k}(\pi _{k}(L)){\mbox{ for all }}1\leq k<n\implies V_{n}(K)\leq V_{n}(L).}

Vk(πk(K)) is sometimes known as the brightness of K and the function Vk o πk 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. 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

Notes

  • Gardner, Richard J. (2002). . Bulletin of the American Mathematical Society. New Series. 39 (3): 355–405 (electronic). doi:.
  • Petty, Clinton M. (1967). "Projection bodies". Proceedings of the Colloquium on Convexity (Copenhagen, 1965). Kobenhavns Univ. Mat. Inst., Copenhagen. pp. 234–241. MR .