Assume that we are given every distance between each pair of points from a $n$-simplex $\triangle$. Given $n$-dimensional barycentric coordinates (measured with respect to $\triangle$) of two points, how do we compute the distance between the two points?
A more detailed explanation of the question is given below:
Given $n+1$ vectors $\{\mathbf{a}_1,\cdots,\mathbf{a}_{n+1}\}$ from $\mathbb{R}^n$, we can give homogeneous coordinates to another vector $\mathbf{x}\in\mathbb{R}^{n+1}$ as $(\lambda x_1:\lambda x_2:\cdots:\lambda x_n)$ where $\lambda\in\mathbb{R}-\{0\}$ and $\mathbf{x}={(\sum x_i \mathbf{a}_i)}/{(\sum x_i)}$. $x_i$ can be alternatively defined as the volume of the simplex formed by vectors $\{\mathbf{x},\mathbf{a}_1, \cdots, \mathbf{a}_{i-1},\mathbf{a}_{i+1},\cdots,\mathbf{a}_{n+1}\}$ (which can be computed via Cayley-Menger determinant, I believe.)
Assume that we are given matrix $M$ such that $M_{ij}=\|\mathbf{a}_i-\mathbf{a}_j\|^2$, i.e. we are given the distances between arbitrary two points from $\{\mathbf{a}_1,\cdots,\mathbf{a}_{n+1}\}$. If we are given barycentric coordinates $P,Q\in\mathbb{R}^{n+1}$ of two vectors $\mathbf{p},\mathbf{q}\in\mathbb{R}^n$, can we express $\|\mathbf{p}-\mathbf{q}\|^2$ in terms of $P,Q,M$?
This is already answered for $n=1,2$. For $n=1$ it is trivial. For $n=2$, assuming that $P$ and $Q$ are normalized (that is, their coordinate sums are equal to 1), and $P=\{p_1:p_2:p_3\},Q=\{q_1:q_2:q_3\}$, distance $\|\mathbf{p}-\mathbf{q}\|^2=\sum_{cyc}\frac12(-a^2+b^2+c^2)(p_1-q_1)^2$ where $a,b,c$ are sidelengths of the simplex.