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[edit]

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.

References[edit]

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

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