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Busemann-Petty Problem


If the section function of a centered convex body in n-dimensional Euclidean space (n>=3) 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 n<=4 and negative if n>=5. This solution appeared as the result of work of many mathematicians; see e.g., Gardner et al. (1999) and Zhang (1999) for historical details.


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Tomography

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References

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. "Comparison of Volumes by Means of the Areas of Central Sections." http://www.math.missouri.edu/~koldobsk/publications/comp.pdf.Koldobsky, A. "A Generalization of the Busemann-Petty Problem on Sections of Convex Bodies." Israel J. Math. 110, 75-91, 1999.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. http://arxiv.org/abs/math.MG/0406406.

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Busemann-Petty Problem

Cite this as:

Weisstein, Eric W. "Busemann-Petty Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Busemann-PettyProblem.html

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