Minkowski's first inequality for convex bodies

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.

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.

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.

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.

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