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':

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

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.

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':

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

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.

I should say why I need to know if this result is true. I am studying an article of Baldo: Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, where he proves that the following functional $$ \mathcal{F}(E_1,...,E_n) = \sum_{1\leq i < j\leq n} d(\alpha_i,\alpha_j) \mathcal{H}^{N-1}(\partial^*E_i \cap \partial^*E_j \cap \Omega) $$ is a $\Gamma$-limit of certain functionals, and therefore it is lower semicontinuous, where $d(\alpha_i,\alpha_j)$ is defined as above. I was wondering if it is possible to prove that for any $\sigma_{ij}$ which satisfy the triangle inequality (which is a necessary physical condition), the lower semicontinuity, and therefore the existence of a minimum for the given energy, $$ \mathcal{F}(E_1,...,E_n) = \sum_{1\leq i < j\leq n} \sigma_{ij} \mathcal{H}^{N-1}(\partial^*E_i \cap \partial^*E_j \cap \Omega) $$ still holds.

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.

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':

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

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.