Mixed volume

Remarks

The left side of clause 2 is a homogeneous polynomial of degree $n$ in $\lambda_1, \dots, \lambda_r$; this is the theorem of Minkowski that makes the definition possible. An explicit formula is $$V(K_1, \dots, K_n) = \frac{1}{n!} \sum_{\emptyset \ne J \subseteq \set{1, \dots, n}} (-1)^{n - \abs{J}} \operatorname{vol}\Big(\sum_{j \in J} K_j\Big).$$ Mixed volumes are nonnegative, invariant under translating each argument, linear in each argument for Minkowski addition with nonnegative scalars, and monotone under inclusion. Also $V(K, \dots, K) = \operatorname{vol}(K)$.

Example: $n = 2$, $K$ a compact convex set of area $A$ and perimeter $L$, and $D$ the closed unit disk. The area of $K + tD$ is $A + L t + \pi t^2$, while clause 2 gives $A + 2 V(K, D)\, t + \pi t^2$. So $V(K, D) = L / 2$.

Non-example: without clause 1 the coefficients are not unique. For $n = 2$ one may move all of the coefficient $2 V(K_1, K_2)$ to the term $(i_1, i_2) = (1, 2)$ and put $0$ at $(2, 1)$, which still satisfies clause 2.