Volume computing of a special polyhedron in n-dimensions space

1. Definition of a special polyhedron spanned by a set of $n$ -dimensions vectors

     Definition 1. The polyhedrons spanned by the $n$ -dimensions vectors of matrix $A_{m}=\left[a_{1},a_{2},\cdots,a_{m}\right]$ and the parameter set with the finite interval [0,1] are defined as

$C_{n}(A_{m})=\left\{ \left.c_{1}a_{1}+c_{2}a_{2}+\cdots+c_{m}a_{m}\right|\forall c_{i}\in[0,1],i=1,2,\cdots,m\right\}$

2. Volume computing of the special polyhedron spanned by $n$ vectors in $n$ -dimensions space

    Lemma 1. The volume of $C_{n}(A_{n})$ in the $n$ -dimension space can be computed as

$V_{n}(C_{n}(A_{n}))=\left|\mathrm{det}(A_{n})\right|$

3. Volume computing of the special polyhedron spanned by $m (m

       Lemma 2. The volume of the $m$ -dimension geometry $C_{n}(A_{m})$ in the $n$ -dimension space can be computed as

$V_{m}(C_{n}(A_{m}))=\sqrt{\mathrm{det}(A_{m}^{T}A_{m})}$

4. Volume computing of the special polyhedron spanned by $m(m>n)$ vectors in $n$ -dimensions space

      Theorem 1. The volume $C_{n}(A_{m})$ of in the $n$ -dimension space can be computed as

$V_{r}(C_{n}(A_{m}))=\sum_{(i_{1},i_{2},\cdots,i_{n})\in\Omega_{1,m}^{n}}\left|\mathrm{det}\left[a_{i_{1}},a_{i_{2}},\cdots,a_{i_{n}}\right]\right|\qquad r=n$

            $=\sum_{(i_{1},i_{2},\cdots,i_{r})\in\Omega_{1,m}^{r}}\left(\mathrm{det}\left[\left[a_{i_{1}},a_{i_{2}},\cdots,a_{i_{r}}\right]^{T}\left[a_{i_{1}},a_{i_{2}},\cdots,a_{i_{r}}\right]\right]\right)^{\frac{1}{2}}\qquad r

where $r=\textrm{rank}\left(A_{m}\right)$ and $\Omega_{1,m}^{n}$ is constituted by the all possible multi-tuple $(i_{1},i_{2},\cdots,i_{n})$ which elements are picked from the set $\{1,2,\cdots,m\}$ and sorted by the values.

      The above theorem is proposed and proven in my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems) and the above two lemmas are quoted from the textbook in linear algebra field.