Volume of polyhedron












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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







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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







    share|cite|improve this question

























      1












      1








      1







      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







      share|cite|improve this question













      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






      share|cite|improve this question













      share|cite|improve this question











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      asked 2 days ago









      MathGirl88MathGirl88

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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.






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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.






            share|cite|improve this 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.






              share|cite|improve this 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.






                share|cite|improve this answer












                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.







                share|cite|improve this answer












                share|cite|improve this answer



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                answered yesterday









                Richard StanleyRichard Stanley

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