Brunn–Minkowski theorem

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 L. A. 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:

\( [ \mu (A + B) ]^{1/n} \geq [\mu (A)]^{1/n} + [\mu (B)]^{1/n}, \)

where A + B denotes the Minkowski sum:

\( A + B := \{\, a + b \in \mathbb{R}^{n} \mid a \in A,\ b \in B \,\}. \)

Remarks

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

\( A \mapsto [\mu (A)]^{1/n} \)

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

\( \left[ \mu (t A + (1 - t) B ) \right]^{1/n} \geq t [ \mu (A) ]^{1/n} + (1 - t) [ \mu (B) ]^{1/n}. \)

For convex sets A and B, the inequality in the theorem is strict for 0 < t < 1 unless A and B are homothetic, i.e. are equal up to translation and dilation.

See also

Isoperimetric inequality
Milman's reverse Brunn–Minkowski inequality
Minkowski–Steiner formula
Prékopa–Leindler inequality
Vitale's random Brunn–Minkowski inequality

References

Brunn, H. (1887). "Über Ovale und Eiflächen". Inaugural Dissertation, München.
Fenchel, Werner; Bonnesen, Tommy (1934). Theorie der konvexen Körper. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Berlin: 1. Verlag von Julius Springer.
Fenchel, Werner; Bonnesen, Tommy (1987). Theory of convex bodies. Moscow, Idaho: L. Boron, C. Christenson and B. Smith. BCS Associates.
Dacorogna, Bernard (2004). Introduction to the Calculus of Variations. London: Imperial College Press. ISBN 1-86094-508-2.
Heinrich Guggenheimer (1977) Applicable Geometry, page 146, Krieger, Huntington ISBN 0-88275-368-1 .
Lyusternik, Lazar A. (1935). "Die Brunn–Minkowskische Ungleichnung für beliebige messbare Mengen". Comptes Rendus (Doklady) de l'académie des Sciences de l'uRSS (Nouvelle Série) III: 55–58.
Minkowski, Hermann (1896). Geometrie der Zahlen. Leipzig: Teubner.
Ruzsa, Imre Z. (1997). "The Brunn–Minkowski inequality and nonconvex sets". Geometriae Dedicata 67 (3). pp. 337–348. doi:10.1023/A:1004958110076. MR 1475877.
Rolf Schneider, Convex bodies: the Brunn–Minkowski theory, Cambridge University Press, Cambridge, 1993.

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