Shephard's problem

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

[math]\displaystyle{ V_{k} (\pi_{k} (K)) \leq V_{k} (\pi_{k} (L)) \mbox{ for all } 1 \leq k \lt n \implies V_{n} (K) \leq V_{n} (L). }[/math]

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. 

• Schneider, Rolf (1967). "Zur einem Problem von Shephard über die Projektionen konvexer Körper" (in German). Mathematische Zeitschrift 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