Volume of polyhedron
$begingroup$
Given the following polyhedron: All the $ntimes n$ matrices $boldsymbol{X}$ with elements $x_{ij}in(0,1)$ such that
$$boldsymbol{X}cdotboldsymbol{1}=boldsymbol{r}, boldsymbol{1}^Tboldsymbol{X}=boldsymbol{c}^T$$
For some given vectors $boldsymbol{r}$ and $boldsymbol{c}$.
Can I calculate the volume of this polyhedron? Can I calculate it's surface?
linear-algebra matrices
asked Jan 7 at 8:59
MathGirl88MathGirl88
1085
$endgroup$
add a comment |
$begingroup$
Given the following polyhedron: All the $ntimes n$ matrices $boldsymbol{X}$ with elements $x_{ij}in(0,1)$ such that
$$boldsymbol{X}cdotboldsymbol{1}=boldsymbol{r}, boldsymbol{1}^Tboldsymbol{X}=boldsymbol{c}^T$$
For some given vectors $boldsymbol{r}$ and $boldsymbol{c}$.
Can I calculate the volume of this polyhedron? Can I calculate it's surface?
linear-algebra matrices
asked Jan 7 at 8:59
MathGirl88MathGirl88
1085
$endgroup$
add a comment |
$begingroup$
Given the following polyhedron: All the $ntimes n$ matrices $boldsymbol{X}$ with elements $x_{ij}in(0,1)$ such that
$$boldsymbol{X}cdotboldsymbol{1}=boldsymbol{r}, boldsymbol{1}^Tboldsymbol{X}=boldsymbol{c}^T$$
For some given vectors $boldsymbol{r}$ and $boldsymbol{c}$.
Can I calculate the volume of this polyhedron? Can I calculate it's surface?
linear-algebra matrices
asked Jan 7 at 8:59
MathGirl88MathGirl88
1085
$endgroup$
Given the following polyhedron: All the $ntimes n$ matrices $boldsymbol{X}$ with elements $x_{ij}in(0,1)$ such that
$$boldsymbol{X}cdotboldsymbol{1}=boldsymbol{r}, boldsymbol{1}^Tboldsymbol{X}=boldsymbol{c}^T$$
For some given vectors $boldsymbol{r}$ and $boldsymbol{c}$.
Can I calculate the volume of this polyhedron? Can I calculate it's surface?
linear-algebra matrices
linear-algebra matrices
asked Jan 7 at 8:59
MathGirl88MathGirl88
1085
asked Jan 7 at 8:59
MathGirl88MathGirl88
1085
asked Jan 7 at 8:59
MathGirl88MathGirl88
1085
asked Jan 7 at 8:59
MathGirl88MathGirl88
1085
asked Jan 7 at 8:59
MathGirl88MathGirl88
1085
1085
add a comment |
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
When $boldsymbol{r}=boldsymbol{c}=(1,1,dots,1)$ the polytope $X$ is the Birkhoff polytope of $ntimes n$ doubly-stochastic matrices. Computing its $(n-1)^2$-dimensional volume is a well-known difficult problem. An answer is given by De Loera, Liu, and Yoshida in http://arxiv.org/abs/math/0701866, but it is quite a complicated formula. For generalizing it to $boldsymbol{r}$ and $boldsymbol{c}$ and to higher dimension, see De Loera's survey at http://arxiv.org/pdf/1307.0124.pdf. For additional information in the case of the Birkhoff polytope, see OEIS A078524, A078525, A037302.
answered Jan 7 at 14:39
Richard StanleyRichard Stanley
28.5k8113186
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
When $boldsymbol{r}=boldsymbol{c}=(1,1,dots,1)$ the polytope $X$ is the Birkhoff polytope of $ntimes n$ doubly-stochastic matrices. Computing its $(n-1)^2$-dimensional volume is a well-known difficult problem. An answer is given by De Loera, Liu, and Yoshida in http://arxiv.org/abs/math/0701866, but it is quite a complicated formula. For generalizing it to $boldsymbol{r}$ and $boldsymbol{c}$ and to higher dimension, see De Loera's survey at http://arxiv.org/pdf/1307.0124.pdf. For additional information in the case of the Birkhoff polytope, see OEIS A078524, A078525, A037302.
answered Jan 7 at 14:39
Richard StanleyRichard Stanley
28.5k8113186
$endgroup$
add a comment |
$begingroup$
When $boldsymbol{r}=boldsymbol{c}=(1,1,dots,1)$ the polytope $X$ is the Birkhoff polytope of $ntimes n$ doubly-stochastic matrices. Computing its $(n-1)^2$-dimensional volume is a well-known difficult problem. An answer is given by De Loera, Liu, and Yoshida in http://arxiv.org/abs/math/0701866, but it is quite a complicated formula. For generalizing it to $boldsymbol{r}$ and $boldsymbol{c}$ and to higher dimension, see De Loera's survey at http://arxiv.org/pdf/1307.0124.pdf. For additional information in the case of the Birkhoff polytope, see OEIS A078524, A078525, A037302.
answered Jan 7 at 14:39
Richard StanleyRichard Stanley
28.5k8113186
$endgroup$
add a comment |
$begingroup$
When $boldsymbol{r}=boldsymbol{c}=(1,1,dots,1)$ the polytope $X$ is the Birkhoff polytope of $ntimes n$ doubly-stochastic matrices. Computing its $(n-1)^2$-dimensional volume is a well-known difficult problem. An answer is given by De Loera, Liu, and Yoshida in http://arxiv.org/abs/math/0701866, but it is quite a complicated formula. For generalizing it to $boldsymbol{r}$ and $boldsymbol{c}$ and to higher dimension, see De Loera's survey at http://arxiv.org/pdf/1307.0124.pdf. For additional information in the case of the Birkhoff polytope, see OEIS A078524, A078525, A037302.
answered Jan 7 at 14:39
Richard StanleyRichard Stanley
28.5k8113186
$endgroup$
When $boldsymbol{r}=boldsymbol{c}=(1,1,dots,1)$ the polytope $X$ is the Birkhoff polytope of $ntimes n$ doubly-stochastic matrices. Computing its $(n-1)^2$-dimensional volume is a well-known difficult problem. An answer is given by De Loera, Liu, and Yoshida in http://arxiv.org/abs/math/0701866, but it is quite a complicated formula. For generalizing it to $boldsymbol{r}$ and $boldsymbol{c}$ and to higher dimension, see De Loera's survey at http://arxiv.org/pdf/1307.0124.pdf. For additional information in the case of the Birkhoff polytope, see OEIS A078524, A078525, A037302.
answered Jan 7 at 14:39
Richard StanleyRichard Stanley
28.5k8113186
answered Jan 7 at 14:39
Richard StanleyRichard Stanley
28.5k8113186
answered Jan 7 at 14:39
Richard StanleyRichard Stanley
28.5k8113186
answered Jan 7 at 14:39
Richard StanleyRichard Stanley
28.5k8113186
28.5k8113186
add a comment |
add a comment |
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