In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to a tuple of convex bodies in R n {\displaystyle \mathbb {R} ^{n}} . This number depends on the size and shape of the bodies, and their relative orientation to each other.
Definition Let K 1 , K 2 , … , K r {\displaystyle K_{1},K_{2},\dots ,K_{r}} be convex bodies in R n {\displaystyle \mathbb {R} ^{n}} and consider the function
f ( λ 1 , … , λ r ) = V o l n ( λ 1 K 1 + ⋯ + λ r K r ) , λ i ≥ 0 , {\displaystyle f(\lambda _{1},\ldots ,\lambda _{r})=\mathrm {Vol} _{n}(\lambda _{1}K_{1}+\cdots +\lambda _{r}K_{r}),\qquad \lambda _{i}\geq 0,}
where Vol n {\displaystyle {\text{Vol}}_{n}} stands for the n {\displaystyle n} -dimensional volume, and its argument is the Minkowski sum of the scaled convex bodies K i {\displaystyle K_{i}} . One can show that f {\displaystyle f} is a homogeneous polynomial of degree n {\displaystyle n} , so can be written as
f ( λ 1 , … , λ r ) = ∑ j 1 , … , j n = 1 r V ( K j 1 , … , K j n ) λ j 1 ⋯ λ j n , {\displaystyle f(\lambda _{1},\ldots ,\lambda _{r})=\sum _{j_{1},\ldots ,j_{n}=1}^{r}V(K_{j_{1}},\ldots ,K_{j_{n}})\lambda _{j_{1}}\cdots \lambda _{j_{n}},}
where the functions V {\displaystyle V} are symmetric. For a particular index function j ∈ { 1 , … , r } n {\displaystyle j\in \{1,\ldots ,r\}^{n}} , the coefficient V ( K j 1 , … , K j n ) {\displaystyle V(K_{j_{1}},\dots ,K_{j_{n}})} is called the mixed volume of K j 1 , … , K j n {\displaystyle K_{j_{1}},\dots ,K_{j_{n}}} .
Properties The mixed volume is uniquely determined by the following three properties:
V ( K , … , K ) = Vol n ( K ) {\displaystyle V(K,\dots ,K)={\text{Vol}}_{n}(K)} ;
V {\displaystyle V} is symmetric in its arguments;
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