Shephard's problem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, Shephard's problem, is the following geometrical question asked by Geoffrey Colin Shephard (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

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.

See also[edit]

References[edit]

  • Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. 
  • Petty, C.M. (1967). "Projection bodies". Proc. Colloquium on Convexity (Copenhagen, 1965). Kobenhavns Univ. Mat. Inst., Copenhagen: 234–241. 
  • Schneider, Rolf (1967). "Zur einem Problem von Shephard über die Projektionen konvexer Körper". Math. Z. (in German). 101: 71–82. doi:10.1007/BF01135693. 
  • Shephard, G. C. (1964), "Shadow systems of convex sets", Israel Journal of Mathematics, 2: 229–236, doi:10.1007/BF02759738, ISSN 0021-2172, MR 0179686