Volume of polyhedron
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 2 days ago
MathGirl88MathGirl88
715
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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 2 days ago
MathGirl88MathGirl88
715
add a comment |
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 2 days ago
MathGirl88MathGirl88
715
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 2 days ago
MathGirl88MathGirl88
715
asked 2 days ago
MathGirl88MathGirl88
715
asked 2 days ago
MathGirl88MathGirl88
715
asked 2 days ago
MathGirl88MathGirl88
715
asked 2 days ago
MathGirl88MathGirl88
715
715
add a comment |
add a comment |
1 Answer
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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 yesterday
Richard StanleyRichard Stanley
28.4k8113185
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1 Answer
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1 Answer
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active
oldest
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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 yesterday
Richard StanleyRichard Stanley
28.4k8113185
add a comment |
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 yesterday
Richard StanleyRichard Stanley
28.4k8113185
add a comment |
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 yesterday
Richard StanleyRichard Stanley
28.4k8113185
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 yesterday
Richard StanleyRichard Stanley
28.4k8113185
answered yesterday
Richard StanleyRichard Stanley
28.4k8113185
answered yesterday
Richard StanleyRichard Stanley
28.4k8113185
answered yesterday
Richard StanleyRichard Stanley
28.4k8113185
28.4k8113185
add a comment |
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