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.

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


 * $$n V_{1} (K, L) = \lim_{\varepsilon \downarrow 0} \frac{V (K + \varepsilon L) - V(K)}{\varepsilon},$$

where V denotes the n-dimensional Lebesgue measure and + denotes the Minkowski sum. Then


 * $$V_{1} (K, L) \geq V(K)^{(n - 1) / n} V(L)^{1 / n},$$

with equality if and only if K and L are homothetic, i.e. are equal up to translation and dilation.

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 &minus; 1)-dimensional surface measure of K, denoted S(K).

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.

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


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

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