Brunn-Minkowski theorem

Let $K_0$ and $K_1$ be convex sets in an $n$-dimensional Euclidean space; let $K_\lambda$, $\lambda\in[0,1]$, be the set of points which divide segments with end points at any points of the sets $K_0$ and $K_1$ in the ratio $\lambda/(1-\lambda)$ (a linear combination of $K_0$ and $K_1$); and let $V(\lambda)$ be the $n$-th power root of the volume of the set $K_\lambda$. Then $V(\lambda)$ is a concave function of $\lambda$, i.e. the inequality

$$V(\lambda_1(1-\rho)+\lambda_2\rho)\geq(1-\rho)V(\lambda_1)+\rho V(\lambda_2)$$

is valid for all $\lambda_1,\lambda_2,\rho\in[0,1]$. The function $V(\lambda)$ is linear (and the inequality then becomes an equality) if and only if $K_0$ and $K_1$ are homothetic. The Brunn–Minkowski theorem can be generalized to linear combinations of several convex sets. It is used to solve extremal and uniqueness problems. It was discovered by H. Brunn in 1887, and completed and rendered more precise in 1897 by H. Minkowski.

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