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 (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).

Contents 1 Statement of the theorem 2 Remarks 3 See also 4 References

[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:

where A + B denotes the Minkowski sum:

[edit] Remarks

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

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

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] See also

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

[edit] 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. 
• 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.