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Brunn-Minkowski inequality

(Theorem)

Let and be non-empty compact subsets of $\mathbb{R}^d$ . Then

$\displaystyle {\mathrm{vol}}(A+B)^{1/d}\geq{\mathrm{vol}}(A)^{1/d}+{\mathrm{vol}}(B)^{1/d},$

where

denotes the Minkowski sum of

and

, and ${\mathrm{vol}}(S)$ denotes the volume of

References

1: Jiří Matoušek.
Lectures on Discrete Geometry, volume 212 of GTM.
Springer, 2002.
Zbl 0999.52006.

"Brunn-Minkowski inequality" is owned by bbukh.

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Cross-references: volume, Minkowski sum, compact subsets
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This is version 4 of Brunn-Minkowski inequality, born on 2003-09-26, modified 2006-07-01.
Object id is 4743, canonical name is BrunnMinkowskiInequality.
Accessed 3810 times total.

Classification:

51M16 (Geometry :: Real and complex geometry :: Inequalities and extremum problems)

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