Let $n$ line sets be $\mathcal{S}\_i=\{a\mathbf{h}\_i:0 \le a \le 1\}$, for $1 \le i \le n$, where $\{\mathbf{h}\_1,\cdots,\mathbf{h}\_n\}$ is a vector group of rank $r$ in the $r$-dimensional Euclidean space. Define the Minkowski sum of two sets as $\mathcal{S}\_1+\mathcal{S}\_2=\{\mathbf{s}\_1+\mathbf{s}\_2:\mathbf{s}\_1\in\mathcal{S}\_1,~\mathbf{s}\_2\in\mathcal{S}\_2\}$. How to compute the $r$-dimensional volume of $\mathcal{S}\_1+\cdots+\mathcal{S}\_n$?



The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets
Let $M$ be the matrix whose rows are the vectors $\boldsymbol{h\_i}$. Then the $r$-dimensional volume of $\mathcal{S}=\mathcal{S}\_1+\cdots+\mathcal{S}\_n$ is equal to the sum of the absolute values of the $r\times r$ minors of $M$. I don't know who originally showed this, but one can show that $\mathcal{S}$ can be tiled with $r$-dimensional parallelotopes whose volumes are the $r\times r$ minors. This follows for instance from the proof of Lemma 2.1 [here](http://math.mit.edu/~rstan/pubs/pubfiles/83.pdf).


