I'm interested in $n \times m$ joint probability tables with prescribed row and column marginals. Such tables form a convex set known as the transportation polytope. What are the extreme points of this set?
For example, for a $2 \times 2$ case of $$\begin{bmatrix} x_{11} & x_{12}\\ x_{21} & x_{22} \end{bmatrix}$$ with row constraint $x_{11} + x_{12} = 0.9$, column constraint $x_{11} + x_{21} = 0.8$, and $\sum_{i,j} x_{ij} = 1$, then there are two extreme points, $$\begin{bmatrix} 0.8 & 0.1\\ 0 & 0.1 \end{bmatrix}, \quad \begin{bmatrix} 0.7 & 0.2\\ 0.1 & 0 \end{bmatrix}.$$ And every joint table with the constraint lies in the convex hull of these two points.