en>David Eppstein: stub sort

2012-09-05T04:17:13Z

stub sort

← Older revision		Revision as of 06:17, 5 September 2012
Line 1:		Line 1:
	~~Wilber Berryhill~~ is the ~~name his parents gave him~~ and ~~he totally digs~~ that ~~title~~. ~~As a woman what she truly likes~~ is ~~style~~ and ~~she~~'s ~~been performing it for quite~~ a ~~while~~. ~~Distributing production has been his occupation for some time~~. ~~North Carolina is~~ the ~~place he loves most but now he~~ is ~~considering other options.~~<br><br>~~Feel free~~ to ~~surf to my web-site~~; ~~free psychic~~ (~~[http:~~//~~black7~~.~~mireene~~.~~com~~/~~aqw/5741 this contact~~ form])		In [[mathematics]], particularly, in asymptotic [[convex geometry]], '''Milman's reverse Brunn–Minkowski inequality''' is a result due to [[Vitali Milman]]<ref>{{harvtxt\|Milman\|1986}}</ref> that provides a reverse inequality to the famous [[Brunn–Minkowski inequality]] for [[convex body\|convex bodies]] in ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup>. Namely, it bounds the volume of the [[Minkowski sum]] of two bodies from above in terms of the volumes of the bodies.

			==Introduction==

			Let ''K'' and ''L'' be convex bodies in '''R'''<sup>''n''</sup>. The Brunn–Minkowski inequality states that

			:<math> \mathrm{vol}(K+L)^{1/n} \geq \mathrm{vol}(K)^{1/n} + \mathrm{vol}(L)^{1/n}~,</math>

			where vol denotes ''n''-dimensional [[Lebesgue measure]] and the + on the left-hand side denotes Minkowski addition.

			In general, no reverse bound is possible, since one can find convex bodies ''K'' and ''L'' of unit volume so that the volume of their Minkowski sum is arbitrarily large. Milman's theorem states that one can replace one of the bodies by its image under a properly chosen volume-preserving [[linear map]] so that the left-hand side of the Brunn–Minkowski inequality is bounded by a constant multiple of the right-hand side.

			The result is one of the main structural theorems in the local theory of [[Banach spaces]].<ref>{{harvtxt\|Pisier\|1989}}</ref>

			==Statement of the inequality==

			There is a constant ''C'', independent of ''n'', such that for any two centrally symmetric convex bodies ''K'' and ''L'' in '''R'''<sup>''n''</sup>, there are volume-preserving linear maps ''φ'' and ''ψ'' from '''R'''<sup>''n''</sup> to itself such that for any real numbers ''s'', ''t'' > 0

			:<math>\mathrm{vol} ( s \, \varphi K + t \, \psi L )^{1/n} \leq C \left( s\, \mathrm{vol} ( \varphi K )^{1/n} + t\, \mathrm{vol} ( \psi L )^{1/n} \right)~.</math>

			One of the maps may be chosen to be the identity.<ref>{{harvtxt\|Pisier\|1989}}</ref>

			==Notes==
			{{Reflist}}

			==References==

			* {{cite journal\|
			mr=0827101\|
			last=Milman\|
			first=Vitali D.\|
			title=Inégalité de Brunn-Minkowski inverse et applications à la théorie locale des espaces normés. [An inverse form of the Brunn-Minkowski inequality, with applications to the local theory of normed spaces] \|
			journal=C. R. Acad. Sci. Paris Sér. I Math.\|
			volume= 302\|
			year=1986\|
			issue=1\|
			pages=25–28\|ref=harv}}

			* {{cite book\|
			mr=1036275\|
			last=Pisier\|
			first= Gilles\|
			title=The volume of convex bodies and Banach space geometry\|
			series=Cambridge Tracts in Mathematics\|volume=94\|
			publisher=Cambridge University Press\|
			location=Cambridge\|
			year=1989\|
			isbn=0-521-36465-5\|ref=harv}}

			{{DEFAULTSORT:Milman's reverse Brunn-Minkowski inequality}}
			[[Category:Euclidean geometry]]
			[[Category:Geometric inequalities]]
			[[Category:Asymptotic geometric analysis]]

en>Helpful Pixie Bot: ISBNs (Build KE)

2012-05-08T04:50:24Z

ISBNs (Build KE)

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Wilber Berryhill is the name his parents gave him and he totally digs that title. As a woman what she truly likes is style and she's been performing it for quite a while. Distributing production has been his occupation for some time. North Carolina is the place he loves most but now he is considering other options.<br><br>Feel free to surf to my web-site; free psychic ([http://black7.mireene.com/aqw/5741 this contact form])

← Older revision		Revision as of 06:17, 5 September 2012
Line 1:		Line 1:
	~~Wilber Berryhill~~ is the ~~name his parents gave him~~ and ~~he totally digs~~ that ~~title~~. ~~As a woman what she truly likes~~ is ~~style~~ and ~~she~~'s ~~been performing it for quite~~ a ~~while~~. ~~Distributing production has been his occupation for some time~~. ~~North Carolina is~~ the ~~place he loves most but now he~~ is ~~considering other options.~~<br><br>~~Feel free~~ to ~~surf to my web-site~~; ~~free psychic~~ (~~[http:~~//~~black7~~.~~mireene~~.~~com~~/~~aqw/5741 this contact~~ form])		In [[mathematics]], particularly, in asymptotic [[convex geometry]], '''Milman's reverse Brunn–Minkowski inequality''' is a result due to [[Vitali Milman]]<ref>{{harvtxt\|Milman\|1986}}</ref> that provides a reverse inequality to the famous [[Brunn–Minkowski inequality]] for [[convex body\|convex bodies]] in ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup>. Namely, it bounds the volume of the [[Minkowski sum]] of two bodies from above in terms of the volumes of the bodies.

			==Introduction==

			Let ''K'' and ''L'' be convex bodies in '''R'''<sup>''n''</sup>. The Brunn–Minkowski inequality states that

			:<math> \mathrm{vol}(K+L)^{1/n} \geq \mathrm{vol}(K)^{1/n} + \mathrm{vol}(L)^{1/n}~,</math>

			where vol denotes ''n''-dimensional [[Lebesgue measure]] and the + on the left-hand side denotes Minkowski addition.

			In general, no reverse bound is possible, since one can find convex bodies ''K'' and ''L'' of unit volume so that the volume of their Minkowski sum is arbitrarily large. Milman's theorem states that one can replace one of the bodies by its image under a properly chosen volume-preserving [[linear map]] so that the left-hand side of the Brunn–Minkowski inequality is bounded by a constant multiple of the right-hand side.

			The result is one of the main structural theorems in the local theory of [[Banach spaces]].<ref>{{harvtxt\|Pisier\|1989}}</ref>

			==Statement of the inequality==

			There is a constant ''C'', independent of ''n'', such that for any two centrally symmetric convex bodies ''K'' and ''L'' in '''R'''<sup>''n''</sup>, there are volume-preserving linear maps ''φ'' and ''ψ'' from '''R'''<sup>''n''</sup> to itself such that for any real numbers ''s'', ''t'' > 0

			:<math>\mathrm{vol} ( s \, \varphi K + t \, \psi L )^{1/n} \leq C \left( s\, \mathrm{vol} ( \varphi K )^{1/n} + t\, \mathrm{vol} ( \psi L )^{1/n} \right)~.</math>

			One of the maps may be chosen to be the identity.<ref>{{harvtxt\|Pisier\|1989}}</ref>

			==Notes==
			{{Reflist}}

			==References==

			* {{cite journal\|
			mr=0827101\|
			last=Milman\|
			first=Vitali D.\|
			title=Inégalité de Brunn-Minkowski inverse et applications à la théorie locale des espaces normés. [An inverse form of the Brunn-Minkowski inequality, with applications to the local theory of normed spaces] \|
			journal=C. R. Acad. Sci. Paris Sér. I Math.\|
			volume= 302\|
			year=1986\|
			issue=1\|
			pages=25–28\|ref=harv}}

			* {{cite book\|
			mr=1036275\|
			last=Pisier\|
			first= Gilles\|
			title=The volume of convex bodies and Banach space geometry\|
			series=Cambridge Tracts in Mathematics\|volume=94\|
			publisher=Cambridge University Press\|
			location=Cambridge\|
			year=1989\|
			isbn=0-521-36465-5\|ref=harv}}

			{{DEFAULTSORT:Milman's reverse Brunn-Minkowski inequality}}
			[[Category:Euclidean geometry]]
			[[Category:Geometric inequalities]]
			[[Category:Asymptotic geometric analysis]]

Positive current - Revision history

en>David Eppstein: stub sort

en>Helpful Pixie Bot: ISBNs (Build KE)