Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros $\alpha_1,...,\alpha_n$. Define the following 'distance':

$$ d(\alpha_i,\alpha_j)=\inf\{\int_0^1 \sqrt{W(\gamma(t))}| \gamma'(t)|dt : \gamma \in C^1([0,1];\Bbb{R}^n),\gamma>0, \gamma(0)=\alpha_i,\ \gamma(1)=\alpha_j\}$$ 

Suppose I have a set of real, positive numbers $\sigma_{ij}>0,\ i \neq j$ with the property that $\sigma_{ij}=\sigma_{ji}$ and $\sigma_{ij} \leq \sigma_{ik}+\sigma_{kj},i,j,k=1,...,n$.

My question is: 

> Can we find $\alpha_i, i=1..n$ and $W$ with the desired properties, such that $d(\alpha_i,\alpha_j)=\sigma_{ij}$? 

I feel that the fact that we can choose $W$ and the zeros of $W$, $\alpha_1,...,\alpha_n$ gives enough freedom for us to solve this system. Thank you.