// Workers AI · dad joke modeWhat did Shephard's problem say to the math book? "You're flocking with the wrong equations.

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 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

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

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 and the notion of projection bodies of convex bodies.

See also

• Busemann–Petty problem

References

• Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bulletin of the American Mathematical Society. New Series. 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.
• Petty, Clinton M. (1967). "Projection bodies". Proceedings of the Colloquium on Convexity (Copenhagen, 1965). Kobenhavns Univ. Mat. Inst., Copenhagen. pp. 234–241. MR 0216369.

• Schneider, Rolf (1967). "Zur einem Problem von Shephard über die Projektionen konvexer Körper". Mathematische Zeitschrift (in German). 101: 71–82. doi:10.1007/BF01135693.
• Shephard, G. C. (1964), "Shadow systems of convex sets", Israel Journal of Mathematics, 2 (4): 229–236, doi:10.1007/BF02759738, ISSN 0021-2172, MR 0179686