What is mixed volume

Minkowski sums of convex bodies

The Minkowski sum of a curve and a two dimensional shape is fairly easy to understand in an intuitive sense.

Mixed volume: the definition

Given \(n\) convex bodies (nonempty convex sets) \(A_1,\ldots,A_n\) in \(\mathbb{R}^n\), we consider their scaled versions \(\lambda_1 A_1, \ldots, \lambda_n A_n\) by positive scaling factors \(\lambda_1,\ldots,\lambda_n\). It can be shown that the Minkowski sum \(\lambda_1 \, A_1 \, + \, \cdots \, + \, \lambda_n \, A_n\) is also a convex body.

is a homogeneous polynomial in the variables \(\lambda_1,\ldots,\lambda_n\). This is the content of the Minkowki Theorem and essence of mixed volume. This polynomial contains almost everything one needs to understand the concept of mixed volume.

The notation

is often used.

Simple case: mixed area

It is perhaps best to start with the simplest case: the mixed area.

A functional point of view

It turns out mixed volume, as a function on several convex bodies, is uniquely characterized by some of its properties:

• MVol is symmetric, i.e., its value is invariant under any permutation of the arguments;
• MVol is additive with respect to the Minkowski sum;
• \(\operatorname{MVol}(A,\ldots,A) = n! \, \operatorname{Vol}_n(A) \)