Shephard's problem

In mathematics, Shephard's problem is the following geometrical question: 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" : R^"n" → Π_"k" is a projection of R^"n" 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"("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.

References

* cite journal
last=Gardner
first=Richard J.
title=The Brunn-Minkowski inequality
journal=Bull. Amer. Math. Soc. (N.S.)
volume=39
issue=3
year=2002
pages=355–405 (electronic)
doi=10.1090/S0273-0979-02-00941-2
* cite journal
last = Petty
first = C.M.
title = Projection bodies
journal = Proc. Colloquium on Convexity (Copenhagen, 1965)
pages = 234–241
publisher = Kobenhavns Univ. Mat. Inst., Copenhagen
year = 1967
* cite journal
last = Schneider
first = Rolf
title = Zur einem Problem von Shephard über die Projektionen konvexer Körper
journal = Math. Z.
volume = 101
year = 1967
pages = 71–82
language = German
doi = 10.1007/BF01135693