Brunn-Minkowski theorem

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In mathematics, the Brunn-Minowski 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 (H. Brunn 1887; H. Minkowski 1896) applied to convex sets; the generalization to compact nonconvex sets stated here is due to L.A. Lyusternik (1935).

[edit] Statement of the theorem

Let n ≥ 1 and let μ denote Lebesgue measure on Rⁿ. Let A and B be two 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:

$A + B := \{ a + b \in \mathbb{R}^{n} | a \in A, b \in B \}.$

[edit] Remarks

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

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

is concave. Thus, for every pair of compact subsets A and B of Rⁿ and every 0 ≤ t ≤ 1,

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

One can even show that the function is strictly concave. This implies that the inequality in the theorem is strict unless A and B are homothetic, i.e. are equal up to translation and dilation.

[edit] References

Brunn, H. (1887). "Über Ovale und Eiflächen". Inaugural Dissertation, München.

Dacorogna, Bernard (2004). Introduction to the Calculus of Variations. London: Imperial College Press. ISBN 1-86094-508-2.

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.