Brunn–Minkowski theorem

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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.