Mixed-volume theory - Encyclopedia of Mathematics

A branch of the theory of convex bodies concerned with the functionals that arise in the study of linear combinations of bodies (see Addition of sets). The volume V 𝑉 of a linear combination ∑ r i=1 λ i K i ∑ 𝑖 = 1 𝑟 𝜆 𝑖 𝐾 𝑖 of convex bodies K i 𝐾 𝑖 in a Euclidean space R n 𝑅 𝑛 with coefficients λ i ≥0 𝜆 𝑖 ≥ 0 is a homogeneous polynomial of degree n 𝑛 in λ 1 … λ r 𝜆 1 … 𝜆 𝑟 : V( ∑ i=1 r λ i K i )=  ∑ i 1 =1 r … ∑ i n =1 r V i 1 … i n λ i 1 … i n . (*) (*) 𝑉 ( ∑ 𝑖 = 1 𝑟 𝜆 𝑖 𝐾 𝑖 ) =   ∑ 𝑖 1 = 1 𝑟 … ∑ 𝑖 𝑛 = 1 𝑟 𝑉 𝑖 1 … 𝑖 𝑛 𝜆 𝑖 1 … 𝑖 𝑛 . The coefficients V i 1 … i n 𝑉 𝑖 1 … 𝑖 𝑛 are assumed to be symmetric with respect to permutations of the subscripts and are denoted by V( K i 1 … K i n ) 𝑉 ( 𝐾 𝑖 1 … 𝐾 𝑖 𝑛 ) , since they depend only on the bodies K i 1 … K i n 𝐾 𝑖 1 … 𝐾 𝑖 𝑛 . These coefficients are called the mixed volumes of the bodies K i 1 … K i n 𝐾 𝑖 1 … 𝐾 𝑖 𝑛 . The significance of this theory lies in the univer

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