Busemann–Petty problem

In the mathematical field of convex geometry, the Busemann–Petty problem, introduced by Herbert Busemann and Clinton Myers Petty (1956, problem 1), asks whether it is true that a symmetric convex body with larger central hyperplane sections has larger volume. More precisely, if K, T are symmetric convex bodies in Rⁿ such that

$\mathrm {Vol} _{n-1}\,(K\cap A)\leq \mathrm {Vol} _{n-1}\,(T\cap A)$

for every hyperplane A passing through the origin, is it true that Vol_n K ≤ Vol_n T?

Busemann and Petty showed that the answer is positive if K is a ball. In general, the answer is positive in dimensions at most 4, and negative in dimensions at least 5.

History

Unexpectedly at the time, Larman and Claude Ambrose Rogers (1975) showed that the Busemann–Petty problem has a negative solution in dimensions at least 12, and this bound was reduced to dimensions at least 5 by several other authors. Ball (1988) pointed out a particularly simple counterexample: all sections of the unit volume cube have measure at most √2, while in dimensions at least 10 all central sections of the unit volume ball have measure at least √2. Lutwak (1988) introduced intersection bodies, and showed that the Busemann–Petty problem has a positive solution in a given dimension if and only if every symmetric convex body is an intersection body. An intersection body is a star body whose radial function in a given direction u is the volume of the hyperplane section u^⊥ ∩ K for some fixed star body K. Gardner (1994) used Lutwak's result to show that the Busemann–Petty problem has a positive solution if the dimension is 3. Zhang (1994) claimed incorrectly that the unit cube in R⁴ is not an intersection body, which would have implied that the Busemann–Petty problem has a negative solution if the dimension is at least 4. However Koldobsky (1998a) showed that a centrally symmetric star-shaped body is an intersection body if and only if the function 1/||x|| is a positive definite distribution, where ||x|| is the homogeneous function of degree 1 that is 1 on the boundary of the body, and Koldobsky (1998b) used this to show that the unit balls l^p
_n, 1 < p ≤ ∞ in n-dimensional space with the l^p norm are intersection bodies for n = 4 but are not intersection bodies for n ≥ 5, showing that Zhang's result was incorrect. Zhang (1999) then showed that the Busemann–Petty problem has a positive solution in dimension 4. Richard J. Gardner, A. Koldobsky, and T. Schlumprecht (1999) gave a uniform solution for all dimensions.

See also

• Shephard's problem

References

• Ball, Keith (1988), "Some remarks on the geometry of convex sets", Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Berlin, New York: Springer-Verlag, pp. 224–231, doi:10.1007/BFb0081743, ISBN 978-3-540-19353-1, MR 0950983
• Busemann, Herbert; Petty, Clinton Myers (1956), "Problems on convex bodies", Mathematica Scandinavica, 4: 88–94, doi:10.7146/math.scand.a-10457, ISSN 0025-5521, MR 0084791, archived from the original on 2011-08-25
• Gardner, Richard J. (1994), "A positive answer to the Busemann-Petty problem in three dimensions", Annals of Mathematics, Second Series, 140 (2): 435–447, doi:10.2307/2118606, ISSN 0003-486X, JSTOR 2118606, MR 1298719
• Gardner, Richard J.; Koldobsky, A.; Schlumprecht, Thomas B. (1999), "An analytic solution to the Busemann-Petty problem on sections of convex bodies", Annals of Mathematics, Second Series, 149 (2): 691–703, arXiv:math/9903200, doi:10.2307/120978, ISSN 0003-486X, JSTOR 120978, MR 1689343
• Koldobsky, Alexander (1998a), "Intersection bodies, positive definite distributions, and the Busemann-Petty problem", American Journal of Mathematics, 120 (4): 827–840, CiteSeerX 10.1.1.610.5349, doi:10.1353/ajm.1998.0030, ISSN 0002-9327, MR 1637955
• Koldobsky, Alexander (1998b), "Intersection bodies in R⁴", Advances in Mathematics, 136 (1): 1–14, doi:10.1006/aima.1998.1718, ISSN 0001-8708, MR 1623669
• Koldobsky, Alexander (2005), Fourier analysis in convex geometry, Mathematical Surveys and Monographs, vol. 116, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3787-0, MR 2132704
• Larman, D. G.; Rogers, C. A. (1975), "The existence of a centrally symmetric convex body with central sections that are unexpectedly small", Mathematika, 22 (2): 164–175, doi:10.1112/S0025579300006033, ISSN 0025-5793, MR 0390914
• Lutwak, Erwin (1988), "Intersection bodies and dual mixed volumes", Advances in Mathematics, 71 (2): 232–261, doi:10.1016/0001-8708(88)90077-1, ISSN 0001-8708, MR 0963487
• Zhang, Gao Yong (1994), "Intersection bodies and the Busemann-Petty inequalities in R⁴", Annals of Mathematics, Second Series, 140 (2): 331–346, doi:10.2307/2118603, ISSN 0003-486X, JSTOR 2118603, MR 1298716, The result in this paper is wrong; see the author's 1999 correction.
• Zhang, Gaoyong (1999), "A positive solution to the Busemann-Petty problem in R⁴", Annals of Mathematics, Second Series, 149 (2): 535–543, doi:10.2307/120974, ISSN 0003-486X, JSTOR 120974, MR 1689339