Brunn-Minkowski inequality

The Brunn-Minkowski inequality or the Brunn and Minkowski theorem , named after the two mathematicians Hermann Brunn and Hermann Minkowski , is a classic theorem in the mathematical subfield of convex geometry . The inequality sets the Lebesgue measure of the Minkowski sum of two compact subsets of the n-dimensional Euclidean space in relation to the Lebesgue measure of these two subsets. It has numerous applications and in particular has the isoperimetric inequality .

Representation of the inequality

In summary, the inequality states the following:

(1) One forms with the Lebesgue measure for two non-empty compact subsets ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle \lambda _{n}}$ ${\displaystyle A,B\subseteq \mathbb {R} ^{n}}$

the set of all of two elements of or formable sums , ${\displaystyle A}$ ${\displaystyle B}$

so applies to the resulting Minkowski sum

${\displaystyle A+B=\{a+b\;\colon \;a\in A\;,\;b\in B\}}$

the inequality

${\displaystyle {\sqrt[{n}]{\lambda _{n}(A+B)}}\geq {\sqrt[{n}]{\lambda _{n}(A)}}+{\sqrt[{n}]{\lambda _{n}(B)}}}$  .

(2) Are beyond and even convex bodies , ${\displaystyle A}$ ${\displaystyle B}$

so for every real number with the inequality applies ${\displaystyle t}$ ${\displaystyle 0\leq t\leq 1}$

${\displaystyle {\sqrt[{n}]{\lambda _{n}(tA+(1-t)B)}}\geq t{\sqrt[{n}]{\lambda _{n}(A)}}+(1-t){\sqrt[{n}]{\lambda _{n}(B)}}}$  .

Explanations and Notes

(a) For two non-empty compact subsets , the Minkowski sum is always a compact subset of the and, in particular, Lebesgue measurable . ${\displaystyle A,B\subseteq \mathbb {R} ^{n}}$ ${\displaystyle A+B}$ ${\displaystyle \mathbb {R} ^{n}}$

(b) For a non-empty compact subset and any real number , the set of the elements of multiplied by is also always a compact subset of and, in particular, Lebesgue measurable. ${\displaystyle A\subseteq \mathbb {R} ^{n}}$ ${\displaystyle t}$ ${\displaystyle tA=\{ta\;\colon \;a\in A\}}$ ${\displaystyle t}$ ${\displaystyle A}$ ${\displaystyle \mathbb {R} ^{n}}$

(c) If one disregards the compactness of the two subsets in (1) and only assumes that both may be Lebesgue measurable, it is generally not even guaranteed that their Minkowski sum represents a Lebesgue measurable subset of the . However, if the external Lebesgue measure is used instead of the Lebesgue measure , the above inequality (1) applies accordingly. Even for any non-empty subsets , the inequality always holds  . ${\displaystyle A,B\subseteq \mathbb {R} ^{n}}$ ${\displaystyle A+B}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \lambda _{n}}$ ${\displaystyle {\lambda _{n}}^{*}}$ ${\displaystyle A,B\subseteq \mathbb {R} ^{n}}$ ${\displaystyle {\sqrt[{n}]{{\lambda _{n}}^{*}(A+B)}}\geq {\sqrt[{n}]{{\lambda _{n}}^{*}(A)}}+{\sqrt[{n}]{{\lambda _{n}}^{*}(B)}}}$

literature

• Yu. D. Burago - VA Zalgaller : Geometric Inequalities (=  The basic teachings of the mathematical sciences in individual representations . Volume 285 ). Springer Verlag, Berlin ( inter alia ) 1988, ISBN 3-540-13615-0 ( MR0936419 ).
• Herbert Federer : Geometric Measure Theory (=  The basic teachings of the mathematical sciences in individual representations with special consideration of the areas of application . Volume 153 ). Springer-Verlag, Berlin / Heidelberg / New York 1969 ( MR0257325 ).
• RJ Gardner: The Brunn-Minkowski inequality . In: Bull. Amer. Math. Soc. (NS) . tape 39 , 2002, p. 355-405 ( ams.org ). MR1898210
• H. Hadwiger : Lectures on content, surface and isoperimetry (=  the basic teachings of the mathematical sciences in individual presentations with special consideration of the areas of application . Volume 93 ). Springer-Verlag, Berlin ( inter alia) 1957 ( MR0102775 ).
• Kurt Leichtweiß : Convex quantities (=  university text ). Springer-Verlag, Berlin / Heidelberg / New York 1980, ISBN 3-540-09071-1 .
• Boris Makarov, Anatoly Podkorytov: Real Analysis: . Measures, Integrals and Applications (=  Universitext ). Springer-Verlag, London (inter alia) 2013, ISBN 978-1-4471-5121-0 ( MR3089088 ).
• Vitali D. Milman , Gideon Schechtman: Asymptotic Theory of Finite Dimensional Normed Spaces (=  Lecture Notes in Mathematics . Volume 1200 ). Springer-Verlag, Berlin ( inter alia ) 1986, ISBN 3-540-16769-2 ( MR0856576 ).
• Frederick A. Valentine: Convex sets (=  BI university paperbacks . Volume 402 / 402a). Bibliographisches Institut , Mannheim 1968 ( MR0226495 ).

Individual evidence

1. ↑ Yu. D. Burago, VA Zalgaller: Geometric Inequalities. 1988, p. 136 ff, p. 146
2. ^ H. Hadwiger: Lectures on content, surface and isoperimetry. 1957, p. 187 ff
3. ↑ Kurt Leichtweiß: Convex sets. 1980, p. 248 ff
4. ↑ Vitali D. Milman, Gideon Schechtman: Asymptotic Theory of Finite Dimensional Normed Spaces. 1986, p. 134 ff, p. 146
5. ↑ Boris Makarov, Anatolij Podkorytov: Real Analysis:… 2013, p. 87 ff
6. ↑ Frederick A. Valentine: Convex sets. 1968, pp. 196-197
7. ^ Herbert Federer: Geometric Measure Theory. 1969, p. 277 ff