If the section function of a centered convex body in -dimensional Euclidean space
()
is smaller than that of another such body, is its volume also smaller?
The solution was completed in the end of the 1990s, and the answer is affirmative if
and negative if .
This solution appeared as the result of work of many mathematicians; see e.g., Gardner
et al. (1999) and Zhang (1999) for historical details.
Bourgain, J. and Zhang, G. "On a Generalization of the Busemann-Petty Problem." In Convex Geometric Analysis (Berkeley, CA,
1996). Cambridge, England: Cambridge University Press, pp. 65-76, 1999.Busemann,
H.; and Petty, C. M. "Problems on Convex Bodies." Math. Scand.4,
88-94, 1956.Gardner, R. J. "Geometric Tomography." Not.
Amer. Math. Soc.42, 422-429, 1995.Gardner, R. J. Geometric
Tomography. New York: Cambridge University Press, 1995.Gardner,
R. J.; Koldobsky, A.; and Schlumprecht, T. "An Analytic Solution to the
Busemann-Petty Problem." Ann. Math.149, 691-703, 1999.Koldobsky,
A. "A Generalization of the Busemann-Petty Problem on Sections of Convex Bodies."
Israel J. Math.110, 75-91, 1999.Koldobsky, A. "Comparison
of Volumes by Means of the Areas of Central Sections." http://www.math.missouri.edu/~koldobsk/publications/comp.pdf.Rubin,
B. and Zhang, G. "Generalizations of the Busemann-Petty Problem for Sections
of Convex Bodies." J. Func. Anal.213, 473-501, 2004.Zhang,
G. "A Positive Answer to the Busemann-Petty Problem in Four Dimensions."
Ann. Math.149, 535-543, 1999.Zvavitch, A. "The Busemann-Petty
Problem for Arbitrary Measures." 21 Jun 2004. https://arxiv.org/abs/math/0406406.
-dimensional Euclidean space
(
)
is smaller than that of another such body, is its volume also smaller?
and negative if 
.
This solution appeared as the result of work of many mathematicians; see e.g., Gardner
et al. (1999) and Zhang (1999) for historical details.
