Define the earth mover's distance as
\begin{equation} EMD(\mathbf{x},\mathbf{y}) = \frac{\sum_{i=1}^m\sum_{j=1}^n f_{ij}d(x_i,y_j)}{\sum_{i=1}^m\sum_{j=1}^n f_{ij}}. \end{equation}
where we define a flow between $\mathbf{x}$ and $\mathbf{y}$ as any matrix of the form $F=(f_{ij}) \in \mathcal{R}^{m \times n}$ and where $d(\cdot)$ is a metric. I'm proving that also EMD is a metric. I have some trouble in proving the identity of indiscernibles, i.e.
$EMD(x,y)=0$ if and only if $x=y$.
I'm in trouble because, if you are familiar with the EMD, if $x=y$ the problem reduces to transport an amount of dirt from itself to itself, which means no transportation at all, so intuitively I think that the flow for any $i,j$ should be equal to zero, but then also $d(\cdot)$ is equal to zero since it is a metric. Does this mean that both the numerator and the denominator are equal to zero? It does not make sense to me, can someone explain how to prove this? Thanks.
Our flow $(f_{i,j})$ describes how much of the mass at $x_i$ gets transported to $y_j$, e.g. if you don't move the pile of dirt at all would not be the zero matrix, but $f_{i,i}=x_i$ and $f_{i,j} = 0$ if $i\ne j$.
So the zero division cannot occur unless you allow $\bf{x}$ to be the zero vector, which would be nonsensical in this setting.