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.

\[ \text{Vol}_n(\lambda_1 \, A_1 \, + \, \cdots \, + \, \lambda_n \, A_n) \]

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

\[ \operatorname{MVol} (A_1, \ldots, A_n) \]

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;
  • \(\operatorname{MVol}(A,\ldots,A) = n! \, \operatorname{Vol}_n(A) \)