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In the mathematical field of [[convex geometry]], the '''Busemann–Petty problem''', introduced by {{harvs|txt|first=Herbert|last=Busemann|authorlink=Herbert Busemann|first2=Clinton Myers|last2=Petty|author2-link=Clinton Myers Petty |year=1956|loc=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 '''R'''<sup>''n''</sup> such that


:<math> \mathrm{Vol}_{n-1} \, (K \cap A) \leq \mathrm{Vol}_{n-1} \, (T \cap A) </math>


for every hyperplane ''A'' passing through the origin, is it true that Vol<sub>''n''</sub>&nbsp;''K''&nbsp;≤&nbsp;Vol<sub>''n''</sub>&nbsp;''T''?
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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&nbsp;5.
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==History==
 
{{harvs|txt|last=Larman|author2-link=Claude Ambrose Rogers|first2=Claude Ambrose |last2=Rogers|year=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. {{harvtxt|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. {{harvtxt|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''<sup>⊥</sup>&nbsp;∩&nbsp;''K'' for some fixed star body ''K''.
{{harvtxt|Gardner|1994}} used Lutwak's result to show that the Busemann–Petty problem has a positive solution if the dimension is 3. {{harvtxt|Zhang|1994}} claimed incorrectly that the unit cube in '''R'''<sup>4</sup> 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 {{harvtxt|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 {{harvtxt|Koldobsky|1998b}} used this to show that the unit balls l{{su|b=''n''|p=''p''}}, 1&nbsp;<&nbsp;''p''&nbsp;≤&nbsp;∞ in ''n''-dimensional space with the [[Lp norm|l<sup>''p''</sup> norm]] are intersection bodies for ''n''&nbsp;=&nbsp;4 but are not intersection bodies for ''n''&nbsp;≥&nbsp;5, showing that Zhang's result was incorrect. {{harvtxt|Zhang|1999}} then showed that the  Busemann–Petty problem has a positive solution in dimension 4.
{{harvs|txt | last1=Gardner | first1=Richard J. | last2=Koldobsky | first2=A. | last3=Schlumprecht | first3=T. | title=An analytic solution to the Busemann-Petty problem on sections of convex bodies | doi=10.2307/120978 | doi=10.2307/120978 | mr=1689343  | year=1999 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=149 | issue=2 | pages=691–703}} gave a uniform solution for all dimensions.
 
==See also==
 
*[[Shephard's problem]]
 
==References==
 
*{{Citation | last1=Ball | first1=Keith | title=Geometric aspects of functional analysis (1986/87) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math. | doi=10.1007/BFb0081743 | mr=950983  | year=1988 | volume=1317 | chapter=Some remarks on the geometry of convex sets | pages=224–231}}
*{{Citation | last1=Busemann | first1=Herbert | last2=Petty | first2= Clinton Myers | title=Problems on convex bodies | url=http://www.mscand.dk/issue.php?year=1956&volume=4&issue= | mr=0084791  | year=1956 | journal=Mathematica Scandinavica | issn=0025-5521 | volume=4 | pages=88–94}}
*{{Citation | last1=Gardner | first1=Richard J. | title=A positive answer to the Busemann-Petty problem in three dimensions | doi=10.2307/2118606 | mr=1298719 | year=1994 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=140 | issue=2 | pages=435–447}}
*{{Citation | last1=Gardner | first1=Richard J. | last2=Koldobsky | first2=A. | last3=Schlumprecht | first3=T. | title=An analytic solution to the Busemann-Petty problem on sections of convex bodies | doi=10.2307/120978 | mr=1689343 | year=1999 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=149 | issue=2 | pages=691–703}}
*{{Citation | last1=Koldobsky | first1=Alexander | title=Intersection bodies, positive definite distributions, and the Busemann-Petty problem | url=http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.4koldobsky.pdf | mr=1637955  | year=1998a | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=120 | issue=4 | pages=827–840}}
*{{Citation | last1=Koldobsky | first1=Alexander | title=Intersection bodies in R⁴ | doi=10.1006/aima.1998.1718 | mr=1623669  | year=1998b | journal=Advances in Mathematics | issn=0001-8708 | volume=136 | issue=1 | pages=1–14}}
*{{Citation | last1=Koldobsky | first1=Alexander | title=Fourier analysis in convex geometry | url=http://books.google.com/books?id=UU25A67LVe0C | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-3787-0 | mr=2132704  | year=2005 | volume=116}}
*{{Citation | last1=Larman | first1=D. G. | last2=Rogers | first2=C. A. | title=The existence of a centrally symmetric convex body with central sections that are unexpectedly small | doi=10.1112/S0025579300006033 | mr=0390914  | year=1975 | journal=Mathematika. A Journal of Pure and Applied Mathematics | issn=0025-5793 | volume=22 | issue=2 | pages=164–175}}
*{{Citation | last1=Lutwak | first1=Erwin | title=Intersection bodies and dual mixed volumes | doi=10.1016/0001-8708(88)90077-1 | mr=963487  | year=1988 | journal=Advances in Mathematics | issn=0001-8708 | volume=71 | issue=2 | pages=232–261}}
*{{Citation | last1=Zhang | first1=Gao Yong | title=Intersection bodies and the Busemann-Petty inequalities in R⁴ | doi=10.2307/2118603 | mr=1298716 |id= The result in this paper is wrong; see the author's 1999 correction. | year=1994 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=140 | issue=2 | pages=331–346}}
*{{Citation | last1=Zhang | first1=Gaoyong | title=A positive solution to the Busemann-Petty problem in R⁴ | doi=10.2307/120974 | mr=1689339  | year=1999 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=149 | issue=2 | pages=535–543}}
 
{{DEFAULTSORT:Busemann-Petty problem}}
[[Category:Convex geometry]]

Latest revision as of 21:56, 4 July 2014


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