Shephard's problem

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 π_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

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

V_k(π_k(K)) is sometimes known as the brightness of K and the function V_k o π_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. The solution of Shephard's problem requires Minkowski's first inequality for convex bodies.

See also

• Busemann–Petty problem

References

• 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