Minkowski's first inequality for convex bodies

From Wikipedia, the free encyclopedia

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.

Contents

[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

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.

[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

\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.

[edit] References