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 (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).

tatement of the theorem

Let "n" ≥ 1 and let "μ" denote Lebesgue measure on R^"n". Let "A" and "B" be two compact subsets of R^"n". 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:

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

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 R^"n" and every 0 ≤ "t" ≤ 1,

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

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.

ee also

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

References

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