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| In [[mathematics]], the '''Brunn–Minkowski theorem''' (or '''Brunn–Minkowski inequality''') is an inequality relating the volumes (or more generally [[Lebesgue measure]]s) of [[compact space|compact]] [[subset]]s of [[Euclidean space]]. The original version of the Brunn–Minkowski theorem ([[Hermann Brunn]] 1887; [[Hermann Minkowski]] 1896) applied to convex sets; the generalization to compact nonconvex sets stated here is due to [[Lazar Lyusternik|L. A. Lyusternik]] (1935).
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| ==Statement of the theorem==
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| Let ''n'' ≥ 1 and let ''μ'' denote the [[Lebesgue measure]] on '''R'''<sup>''n''</sup>. Let ''A'' and ''B'' be two nonempty compact subsets of '''R'''<sup>''n''</sup>. Then the following [[inequality (mathematics)|inequality]] holds:
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| :<math>[ \mu (A + B) ]^{1/n} \geq [\mu (A)]^{1/n} + [\mu (B)]^{1/n},</math> | |
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| where ''A'' + ''B'' denotes the [[Minkowski sum]]:
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| :<math>A + B := \{\, a + b \in \mathbb{R}^{n} \mid a \in A,\ b \in B \,\}.</math>
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| ==Remarks==
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| The proof of the Brunn–Minkowski theorem establishes that the function
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| :<math>A \mapsto [\mu (A)]^{1/n}</math>
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| is [[concave function|concave]] in the sense that, for every pair of nonempty compact subsets ''A'' and ''B'' of '''R'''<sup>''n''</sup> and every 0 ≤ ''t'' ≤ 1,
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| :<math>\left[ \mu (t A + (1 - t) B ) \right]^{1/n} \geq t [ \mu (A) ]^{1/n} + (1 - t) [ \mu (B) ]^{1/n}.</math>
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| For [[convex set|convex]] sets ''A'' and ''B'', the inequality in the theorem is strict
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| for 0 < ''t'' < 1 unless ''A'' and ''B'' are [[homothetic]], i.e. are equal up to [[translation (geometry)|translation]] and [[Scaling (geometry)|dilation]].
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| ==See also==
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| * [[Isoperimetric inequality]]
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| * [[Milman's reverse Brunn–Minkowski inequality]]
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| * [[Minkowski–Steiner formula]]
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| * [[Prékopa–Leindler inequality]]
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| * [[Vitale's random Brunn–Minkowski inequality]]
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| ==References==
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| * {{cite journal | author=Brunn, H. | author-link=Hermann Brunn | title=Über Ovale und Eiflächen | year = 1887 | version=Inaugural Dissertation, München}}
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| *{{cite book
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| | last=Fenchel
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| | first=Werner
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| | author-link = Werner Fenchel
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| | coauthors=Bonnesen, Tommy
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| | title=Theorie der konvexen Körper
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| | series=Ergebnisse der Mathematik und ihrer Grenzgebiete
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| | volume=3
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| | publisher=1. Verlag von Julius Springer
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| | location=Berlin
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| | year=1934
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| }}
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| *{{cite book
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| | last=Fenchel
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| | first=Werner
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| | author-link=Werner Fenchel
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| | coauthors=Bonnesen, Tommy
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| | title=Theory of convex bodies
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| | publisher=L. Boron, C. Christenson and B. Smith. BCS Associates
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| | location=Moscow, Idaho
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| | year=1987
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| }}
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| * {{cite book | last=Dacorogna | first=Bernard | title=Introduction to the Calculus of Variations | publisher=Imperial College Press | location=London | year=2004 | isbn=1-86094-508-2 | unused_data=|ISBN status=May be invalid – please double check }}
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| * [[Heinrich Guggenheimer]] (1977) ''Applicable Geometry'', page 146, Krieger, Huntington ISBN 0-88275-368-1 .
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| * {{cite journal | last=Lyusternik | first=Lazar A. | authorlink=Lazar Lyusternik | title=Die Brunn–Minkowskische Ungleichnung für beliebige messbare Mengen | journal = Comptes Rendus (Doklady) de l'académie des Sciences de l'uRSS (Nouvelle Série) | volume = III | year = 1935 | pages = 55–58}}
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| * {{cite book | last=Minkowski | first=Hermann | authorlink=Hermann Minkowski | title = Geometrie der Zahlen | location = Leipzig | publisher = Teubner | year = 1896}}
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| * {{cite article|last=Ruzsa|first=Imre Z.|authorlink=Imre Z. Ruzsa|title=The Brunn–Minkowski inequality and nonconvex sets|journal=Geometriae Dedicata|volume=67|doi=10.1023/A:1004958110076 |doi=10.1023/A:1004958110076|year=1997|number=3|pages=337–348|mr=1475877}}
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| * Rolf Schneider, ''Convex bodies: the Brunn–Minkowski theory,'' Cambridge University Press, Cambridge, 1993.
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| {{DEFAULTSORT:Brunn-Minkowski theorem}}
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| [[Category:Theorems in measure theory]]
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| [[Category:Theorems in convex geometry]]
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| [[Category:Calculus of variations]]
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| [[Category:Geometric inequalities]]
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| [[Category:Sumsets]]
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