Minkowski's first inequality for convex bodies

• 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).

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

${\displaystyle \left({\frac {V(K)}{V(B)}}\right)^{1/n}\leq \left({\frac {S(K)}{S(B)}}\right)^{1/(n-1)},}$

with equality if and only if K is a ball of some radius.

• Gardner, Richard J. (2002). "The Brunn–Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.

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