Angular resolution (graph drawing) - Revision history

en>David Eppstein: Petra Mutzel

2014-07-04T19:56:04Z

← Older revision		Revision as of 21:56, 4 July 2014
Line 1:		Line 1:
	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> ~~'~~'K'' ≤ Vol~~<~~sub~~>~~''n''~~<~~/sub~~>~~ ''T''?~~		Do you have a heart for customer service? If so, you can make money online answering phone calls for businesses. Additionally, you can make money by chatting online with a business's customer to help them solve problems and answer their questions. There are many customer como ganhar dinheiro service sites available including: LiveOps, Working Solutions [http://www.comoganhardinheiro101.com/?p=12 ganhar dinheiro na internet] and ACD Direct. Write online to make money. There are quite a few legitimate companies that pay people to write articles online. You can find many different companies that will pay you to write different content with varying topics. If you enjoy writing, you should do research to look for companies that will pay you to write online. <br><br><br>If you have the ability to type quickly, you may consider transcription jobs. You need to have good headphones and the ability to keep up with what is being said. Often, the companies will have their own formatting guidelines. Familiarize yourself with these prior to beginning. Once you have, you can begin transcribing. [http://www.comoganhardinheiro101.com/?p=71 como ganhar dinheiro] What do you do? Do it for pay! A number of websites offer you points when you search through them and/or make purchases of items that you were planning to buy anyway. Take the points you earn and trade them for gift cards, gifts and sometimes even cash. Sites thatprovide this opportunity include Swagbucks and Mypoints.<br><br>As you can see from the above article, it is easy to understand how to make money online when you get the knowledge through great tips. You can join the millions of people worldwide who make money each day, if you [http://www.comoganhardinheiro101.com/ganhar-dinheiro-pela-internet-com-opcoes-binarias/ como ganhar dinheiro pela internet] pay close attention to the tips that you have just read. Making money online is a skill, and if you continue to study good tips and ideas such as these, there is no reason why you cannot make lots of money online.<br><br>Don't buy every book about making money online. Most of the time, these books are sold by people whose claim to fame is that they are teaching people how to make money [http://www.comoganhardinheiro101.com/inicio/ ganhe dinheiro na internet] online. Before you buy an e-book or book about making money, make sure that you know a little about the author, and that you ganhe dinheiro have seen testimonials about the book.<br><br>Take advantage of paid per post if you have extra time and need to make money online. People will need you to make posts on their site to promote a product or give a good review, which will help them in the long run. In return, they will pay you a nice sum. Do you enjoy [http://www.comoganhardinheiro101.com/ferramentas/ como ganhar dinheiro na internet] writing? If so, you can make cash by becoming a freelance writer. Just be aware that most freelance writing sites prefer writers with experience. But, there are other sites that will hire you if you pass a test. It's a bonus if you have knowledge on particular subjects. <br><br><br>Many sites require a payment for information that they claim will allow you to make a lot of money online. A good rule of thumb is to never pay anything to get that type of information. Legitimate sites may ask you to qualify by taking a test, but they will not require any type of payment. [http://www.comoganhardinheiro101.com/perguntas-frequentes/ como ganhar dinheiro] People love handmade goods. They are very trendy right now, so if you are artsy, go make some money! So, if you are the creative type, whether it's sewing, knitting or any other craft work, now is the time to produce your goods. Both [http://ganhardinheiro.comoganhardinheiro101.com como ganhar dinheiro na internet] Etsy and eBay could be generating an income for you.<br><br>If you want to do something creative, consider ways to create viral videos. Brainstorm ideas that you think are rip [http://Www.comoganhardinheiro101.com/como-ganhar-dinheiro-pela-internet/ riotously] hilarious, then [http://Comoconseguirdinheiro.Comoganhardinheiro101.com/ start putting] them into action. Post them on YouTube, turn on ads and, if they work, watch the money start rolling in! Learn about marketing to see how you can get them even more popular. [http://www.comoganhardinheiro101.com/?p=73 como ganhar dinheiro na internet] If you know how to prepare taxes, you can earn money online. Apply for the necessary government licenses and then begin offering your services online. Many of today's top producers work from home with only their laptop and an email address. For best results, continually advertise your services throughout the year.<br><br>

	~~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~~.		Affiliate marketing can be a good way to earn money online if you have a lot of time and energy to spend on website setup and promotion. Affiliate marketers market the products of others in a number of ways. You can create a website and promote [http://www.comoganhardinheiro101.com/inicio/ como conseguir dinheiro] it. Creating a blog or posting on the blogs of others in your niche is also an excellent way to share product information. Making smart use of social media and e mail marketing are also possibilities for successful affiliate marketing.<br><br>To really make money online, consider setting up your own site and having all the control. You could sell things as an affiliate or offer professional services, among many other things. While working for others can earn you a nice income, there's nothing like raking in all the profits and calling [http://www.comoganhardinheiro101.com/?p=16 como ficar rico] all the shots yourself! If you are you looking for more about [http://comoganhardinheironainternet.comoganhardinheiro101.com ganhar dinheiro] stop by comoganhardinheironainternet.comoganhardinheiro101.com As stated above, there are lots of ways to increase your online income. Use these suggestions to help you get started. You [http://comoganhardinheiropelainternet.comoganhardinheiro101.com como ganhar dinheiro na internet] will be amazed at how quickly the money can add up. Come back to this page and others like it to continue growing your online work portfolio as you go.<br><br>Take advantage of paid per post if you have extra time and need to make money online. People will need you to make posts on their site to promote a product or give a good review, which will help them in the long run. In return, they will pay you a nice sum. Do you enjoy [http://www.comoganhardinheiro101.com/slide-central/ ganhe dinheiro] writing? If so, you can make cash by becoming a freelance writer. Just be aware that most freelance writing sites prefer writers with experience. But, there are other sites that will hire you if you pass a test. It's a bonus if you have knowledge on particular subjects.

	~~==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> ∩ ''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 < ''p'' ≤ ∞ in ''n''-dimensional space with the [[Lp norm\|l<sup>''p''</sup> norm]] are intersection bodies for ~~''n'' = 4 but are not intersection bodies for ''n'' ≥ 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]]~~

en>David Eppstein: /* Relation to graph coloring */ link square of a graph

2013-03-25T03:48:54Z

Relation to graph coloring: link square of a graph

← Older revision		Revision as of 05:48, 25 March 2013
Line 1:		Line 1:
	~~Paulita Rocco~~ is ~~title she would rather be called~~ with and ~~she or he believes suggesting quite sound~~. ~~As anyone what he really likes~~ is ~~[http~~:~~//www~~.~~Answers~~.~~com~~/~~topic/playing+mah playing mah] jongg and He~~'s ~~trying~~ to ~~create it~~ a ~~profession~~. ~~Colorado~~ is ~~just place she~~ has ~~been residing~~ in ~~and she~~'s ~~everything~~ that ~~they needs now there are~~. ~~Her job~~ a ~~auditing police~~ [http://~~Search~~.Un.~~org~~/~~search~~?ie=~~utf8~~&~~site~~=~~un_org~~&~~output~~=~~xml_no_dtd&client~~=~~UN_Website_en&num~~=10~~&lr~~=~~lang_en&proxystylesheet~~=~~UN_Website_en&oe~~=~~utf8&q~~=~~officer&Submit~~=~~Go officer]~~. Go to ~~her site find out more~~: ~~https~~://~~www~~.~~5goal~~.com/~~วิธีเล่นsbobet~~.~~html<br><br>my webpage~~ - ~~[https:~~//~~www~~.~~5goal~~.~~com~~/~~วิธีเล่นsbobet~~.~~html วิธีเล่น sbobet~~]		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> ''K'' ≤ Vol<sub>''n''</sub> ''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==

			{{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> ∩ ''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 < ''p'' ≤ ∞ in ''n''-dimensional space with the [[Lp norm\|l<sup>''p''</sup> norm]] are intersection bodies for ''n'' = 4 but are not intersection bodies for ''n'' ≥ 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]]

en>Jason Davies: /* Outerplanar graphs */ Fix typo.

2012-05-17T23:21:49Z

Outerplanar graphs: Fix typo.

New page

Paulita Rocco is title she would rather be called with and she or he believes suggesting quite sound. As anyone what he really likes is [http://www.Answers.com/topic/playing+mah playing mah] jongg and He's trying to create it a profession. Colorado is just place she has been residing in and she's everything that they needs now there are. Her job a auditing police [http://Search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=officer&Submit=Go officer]. Go to her site find out more: https://www.5goal.com/วิธีเล่นsbobet.html<br><br>my webpage - [https://www.5goal.com/วิธีเล่นsbobet.html วิธีเล่น sbobet]

← Older revision		Revision as of 05:48, 25 March 2013
Line 1:		Line 1:
	~~Paulita Rocco~~ is ~~title she would rather be called~~ with and ~~she or he believes suggesting quite sound~~. ~~As anyone what he really likes~~ is ~~[http~~:~~//www~~.~~Answers~~.~~com~~/~~topic/playing+mah playing mah] jongg and He~~'s ~~trying~~ to ~~create it~~ a ~~profession~~. ~~Colorado~~ is ~~just place she~~ has ~~been residing~~ in ~~and she~~'s ~~everything~~ that ~~they needs now there are~~. ~~Her job~~ a ~~auditing police~~ [http://~~Search~~.Un.~~org~~/~~search~~?ie=~~utf8~~&~~site~~=~~un_org~~&~~output~~=~~xml_no_dtd&client~~=~~UN_Website_en&num~~=10~~&lr~~=~~lang_en&proxystylesheet~~=~~UN_Website_en&oe~~=~~utf8&q~~=~~officer&Submit~~=~~Go officer]~~. Go to ~~her site find out more~~: ~~https~~://~~www~~.~~5goal~~.com/~~วิธีเล่นsbobet~~.~~html<br><br>my webpage~~ - ~~[https:~~//~~www~~.~~5goal~~.~~com~~/~~วิธีเล่นsbobet~~.~~html วิธีเล่น sbobet~~]		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> ''K'' ≤ Vol<sub>''n''</sub> ''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==

			{{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> ∩ ''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 < ''p'' ≤ ∞ in ''n''-dimensional space with the [[Lp norm\|l<sup>''p''</sup> norm]] are intersection bodies for ''n'' = 4 but are not intersection bodies for ''n'' ≥ 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]]