Measurements of a centered convex body in Euclidean n-space (for 
)
show that its brightness function (the volume of each
projection) is smaller than that of another such body. Is it true that its volume
is also smaller? C. M. Petty and R. Schneider showed in 1967 that
the answer is yes if the body with the larger brightness function is a projection
body, but no in general for every 
.
Shephard's Problem
See also
Aleksandrov's Uniqueness Theorem, Brightness, TomographyExplore with Wolfram|Alpha
References
Gardner, R. J. "Geometric Tomography." Not. Amer. Math. Soc. 42, 422-429, 1995.Petty, C. M. "Projection Bodies." In Proc. Colloquium on Convexity (Copenhagen, 1965). Copenhagen, Denmark: Københavns Univ. Mat. Inst., pp. 234-241, 1967.Schneider, R. "Zur einem Problem von Shephard über die Projektionen konvexer Körper." Math. Z. 101, 71-82, 1967.Referenced on Wolfram|Alpha
Shephard's ProblemCite this as:
Weisstein, Eric W. "Shephard's Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ShephardsProblem.html