View source for Busemann–Petty problem

{{Short description|Problem in convex geometry}}
In the mathematical field of [[Convex geometry|convex geometry]], the '''Busemann–Petty problem''', introduced by {{harvs|txt|first=Herbert|last=Busemann|first2=Clinton Myers|last2=Petty|year=1956|loc=problem 1}}, asks whether it is true that a symmetric [[Convex body|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''?

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.

==History==
{{harvs|txt|last=Larman|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 {{radic|2}}, while in dimensions at least 10 all central sections of the unit volume ball have measure at least {{radic|2}}. {{harvs|txt|last=Lutwak|year=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 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. {{harvs|txt|last=Zhang|year=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 | mr=1689343  | year=1999 | journal=[[Annals of Mathematics]] |series=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=[[Physics:Springer-Verlag|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| isbn=978-3-540-19353-1 }}
*{{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 | url-status=dead | archiveurl=https://web.archive.org/web/20110825215028/http://www.mscand.dk/issue.php?year=1956&volume=4&issue= | archivedate=2011-08-25 | doi=10.7146/math.scand.a-10457 | doi-access=free }}
*{{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]] |series=Second Series | issn=0003-486X | volume=140 | issue=2 | pages=435–447| jstor=2118606 }}
*{{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]] |series=Second Series | issn=0003-486X | volume=149 | issue=2 | pages=691–703| arxiv=math/9903200 | jstor=120978 }}
*{{Citation | last1=Koldobsky | first1=Alexander | title=Intersection bodies, positive definite distributions, and the Busemann-Petty problem | mr=1637955  | year=1998a | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=120 | issue=4 | pages=827–840 | doi=10.1353/ajm.1998.0030| citeseerx=10.1.1.610.5349 }}
*{{Citation | last1=Koldobsky | first1=Alexander | title=Intersection bodies in R⁴ | doi=10.1006/aima.1998.1718 | doi-access=free | 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=https://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]] | 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 | doi-access= | 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]] |series=Second Series | issn=0003-486X | volume=140 | issue=2 | pages=331–346| jstor=2118603 }}
*{{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]] |series=Second Series | issn=0003-486X | volume=149 | issue=2 | pages=535–543| jstor=120974 }}

{{DEFAULTSORT:Busemann-Petty problem}}
[[Category:Convex geometry]]

{{Sourceattribution|Busemann–Petty problem}}