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