Mixed volume

Summary

Let $K_1, \dots, K_r$ be nonempty compact convex sets of $\R^n$ and $\lambda_i \ge 0$. The volume of the Minkowski sum $\lambda_1 K_1 + \cdots + \lambda_r K_r$ is a homogeneous polynomial of degree $n$ in the $\lambda_i$. Its coefficients are the mixed volumes $V(K_{i_1}, \dots, K_{i_n})$. Mixed volumes unify volume, surface area, mean width and many other measurements of convex bodies. They satisfy deep inequalities, above all the Alexandrov–Fenchel inequality.