Mixed volume - Knowlpedia

Definition

For convex bodies $K_1,\ldots,K_m\subseteq\mathbb R^m$ , their mixed volume is

V(K_1,\ldots,K_m) =\frac1{m!}[t_1\cdots t_m]\, \operatorname{vol}(t_1K_1+\cdots+t_mK_m).

Here $[t_1\cdots t_m]$ extracts the indicated coefficient from the homogeneous degree- $m$ volume polynomial.

Properties

Mixed volume is symmetric, translation-invariant, and multilinear with respect to Minkowski addition and nonnegative scalar multiplication. It is monotone in each argument and normalized by

V(K,\ldots,K)=\operatorname{vol}(K).

Valuations

Fixing $A_1,\ldots,A_{m-k}$ makes

K\longmapsto V(K[k],A_1,\ldots,A_{m-k})

a continuous translation-invariant valuation homogeneous of degree $k$ . Mixed volume is the convex-body counterpart of determinant polarization, but it is not the same object as the mixed discriminant.

References

Rolf Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd ed., Cambridge University Press, 2014. DOI record. Relevant: Chapter 5.