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Let $H = \{ (a,b,c) \in \mathbb{Z}_{\geq 0}^3 : a+b+c=n \}$. If we have a probability distribution on $H$, we can take its marginals onto the $a$, $b$ and $c$ variables and obtain three probability distributions $\alpha$, $\beta$ and $\gamma$ in $\mathbb{R}^{n+1}$. The set of possible $\alpha$, $\beta$ and $\gamma$ is a polytope in $\mathbb{R}^{3n+3}$. Does anyone know a list of defining inequalities for it?

For those who don't like the probability language: Let $p_{ijk} \in \mathbb{R}_{\geq 0}^H$ with $\sum p_{ijk}=1$. Set $\alpha_i = \sum_{j} p_{ij(n-i-j)}$, $\beta_j = \sum_k p_{(n-j-k)jk}$ and $\gamma_k = \sum_i p_{i(n-i-k)k}$. What are the inequalities defining the possible vectors $\alpha$, $\beta$ and $\gamma$?

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  • $\begingroup$ The polytope in question (call it $P_n$) is the convex hull of $e_a + f_b + g_c$, where $e_0,\dots,e_n,f_0,\dots,f_n,g_0,\dots,g_n$ is a basis for ${\mathbb R}^{3n+3}$ and $(a,b,c)$ ranges over $H$. There appear to be some recursive relations that allow one to describe $P_{n+1}$ or $P_{2n}$ in terms of $P_n$ which may be helpful. $\endgroup$
    – Terry Tao
    Jun 7, 2016 at 16:59
  • $\begingroup$ The defining inequalities correspond to weights $r_0,\dots,r_n,s_0,\dots,s_n,t_0,\dots,t_n$ such that $r_a + s_b + t_c \geq 0$ for all $(a,b,c) \in H$, with equality holding in some "spanning" set of $H$. This looks somewhat combinatorially unappetising, though. $\endgroup$
    – Terry Tao
    Jun 7, 2016 at 17:02
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    $\begingroup$ For n=2. If the variables are $\alpha_0, \alpha_1, \alpha_2, \beta_0, \beta_1, \beta_2, \gamma_0, \gamma_1, \gamma_2$, then the polytope is defined by 6 inequalities and 4 equalities:$$0\le \alpha_2,\beta_2,\gamma_2,d,e,f;$$ $$\alpha_1=e+f;\ \ \beta_1=d+f;\ \ \gamma_1=d+e;$$ $$\alpha_0+\beta_0+\gamma_0=\alpha_2+\beta_2+\gamma_2+1$$ where $d,e,f$ abbreviate $\alpha_0-\beta_2-\gamma_2$, $\beta_0-\alpha_2-\gamma_2$, $\gamma_0-\alpha_2-\beta_2$. $\endgroup$
    – user44143
    Jan 15, 2018 at 5:18

1 Answer 1

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While the literal question I asked is still open, Luke Pebody has proven an important relevant result: If $\alpha_0 \geq \alpha_1 \geq \cdots \geq \alpha_n$, and the same for $\beta$ and $\gamma$, and $\sum \alpha_j + \sum \beta_j + \sum \gamma_k = 3$, then there is a probability distribution $\pi$ with marginals $\alpha$, $\beta$, $\gamma$. This constructs a large polytope inside the one which this question asks for a characterization of.

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