Minkowski's first inequality for convex bodies
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In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely related to the Brunn-Minkowski inequality and the isoperimetric inequality.
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[edit] Statement of the inequality
Let K and L be two n-dimensional convex bodies in n-dimensional Euclidean space Rn. Define a quantity V1(K, L) by
where V denotes the n-dimensional Lebesgue measure and + denotes the Minkowski sum. Then
with equality if and only if K and L are homothetic, i.e. are equal up to translation and dilation.
[edit] Remarks
- V1 is just one example of a class of quantities known as mixed volumes.
- If L is the n-dimensional unit ball B, then n V1(K, B) is the (n − 1)-dimensional surface measure of K, denoted S(K).
[edit] Connection to other inequalties
[edit] The Brunn-Minkowski inequality
One can show that the Brunn-Minkowski inequality for convex bodies in Rn implies Minkowski's first inequality for convex bodies in Rn, and that equality in the Brunn-Minkowski inequality implies equality in Minkowski's first inequality.
[edit] The isoperimetric inequality
By taking L = B, the n-dimensional unit ball, in Minkowski's first inequality for convex bodies, one obtains the isoperimetric inequality for convex bodies in Rn: if K is a convex body in Rn, then
with equality if and only if K is a ball of some radius.
[edit] References
- Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.) 39 (3): 355–405 (electronic). doi: . ISSN 0273-0979.