“Brunn–Minkowski theorem”的意思、由来-开放百科全书

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Brunn–Minkowski theorem

释义

Statement of the theorem
Remarks
See also
References

In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets 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 (1935).

Statement of the theorem

Let n ≥ 1 and let μ denote the Lebesgue measure on Rⁿ. Let A and B be two nonempty compact subsets of Rⁿ. Then the following inequality holds:

where A + B denotes the Minkowski sum:

Remarks

The proof of the Brunn–Minkowski theorem establishes that the function

is concave in the sense that, for every pair of nonempty compact subsets A and B of Rⁿ and every 0 ≤ t ≤ 1,

For convex sets A and B of positive measure, the inequality in the theorem is strict

for 0 < t < 1 unless A and B are positive homothetic, i.e. are equal up to translation and dilation by a positive factor.

References

{{cite journal | author=Brunn, H. | author-link=Hermann Brunn | title=Über Ovale und Eiflächen | year = 1887 | version=Inaugural Dissertation, München}}
{{cite book

| last=Fenchel
| first=Werner
| author-link = Werner Fenchel
|author2=Bonnesen, Tommy
| title=Theorie der konvexen Körper
| series=Ergebnisse der Mathematik und ihrer Grenzgebiete
| volume=3
| publisher=1. Verlag von Julius Springer
| location=Berlin
| year=1934
}}

{{cite book

| last=Fenchel
| first=Werner
| author-link=Werner Fenchel
|author2=Bonnesen, Tommy
| title=Theory of convex bodies
| publisher=L. Boron, C. Christenson and B. Smith. BCS Associates
| location=Moscow, Idaho
| year=1987
}}

{{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}}
Heinrich Guggenheimer (1977) Applicable Geometry, page 146, Krieger, Huntington {{ISBN|0-88275-368-1}} .
{{cite journal | last=Lyusternik | first=Lazar A. | authorlink=Lazar Lyusternik | title=Die Brunn–Minkowskische Ungleichnung für beliebige messbare Mengen | journal = Comptes Rendus de l'Académie des Sciences de l'URSS |series=Nouvelle Série | volume = III | year = 1935 | pages = 55–58}}
{{cite book | last=Minkowski | first=Hermann | authorlink=Hermann Minkowski | title = Geometrie der Zahlen | location = Leipzig | publisher = Teubner | year = 1896}}
{{cite news|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|year=1997|number=3|pages=337–348|mr=1475877}}
Rolf Schneider, Convex bodies: the Brunn–Minkowski theory, Cambridge University Press, Cambridge, 1993.

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Statement of the theorem

Remarks

See also

References