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
\( {\displaystyle V_{k}(\pi _{k}(K))\leq V_{k}(\pi _{k}(L)){\mbox{ for all }}1\leq k<n\implies V_{n}(K)\leq V_{n}(L).} \)
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
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): 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 (4): 229–236, doi:10.1007/BF02759738, ISSN 0021-2172, MR 0179686
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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