Let $\vec{\mu}\_1, \vec{\mu}\_2,\ldots, \vec{\mu}\_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where 
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|\_1 = 1,\mu\_j \geq 0\}.$$
(Here $\mu\_j$ means the $j$-th entry of $\vec{\mu} \in R^d$.) We define $\Delta^{k-1}$ similarly. 


Our goal is to find a $\vec{\mu}^\* \in \Delta^{d-1}$ that satisfies $\vec{\mu}^\* \notin \{\vec{\mu}\_1,\ldots, \vec{\mu}\_k\}$, such that for some $\vec{w} \in \Delta^{k-1}$,  

$$
\sum\_{i} w\_i \vec{\mu}\_i = \vec{\mu}^\*,
$$
and 
$$
\sum\_{i} w\_i \vec{\mu}\_i^{\otimes 2} = (\vec{\mu}^\*)^{\otimes 2}.
$$
(Here $\vec{\mu}\_i^{\otimes 2}$ means $\vec{\mu}\_i\vec{\mu}\_i^\top$.) Equivalently, one can think of constructing $\vec{\mu}\_1, \vec{\mu}\_2,\ldots, \vec{\mu}\_k$ for a given $\vec{\mu}^\*$.


Here is the question: Is there a principled way to do this construction? Or does there exist a solution at all? 


In the simplest example where $k=2$ and $\vec{\mu}^\* = (1/d,\ldots,1/d)^\top$, it seems that the solution doesn't exist. So what about general settings?


**A variant of this problem** is that we want to match
$$
\sum\_{i} w\_i \vec{\mu}\_i = \sum\_{i} w\_i^\* \vec{\mu}\_i^\*,
$$
and 
$$
\sum\_{i} w\_i \vec{\mu}\_i^{\otimes 2} = \sum\_{i} w\_i^\* (\vec{\mu}\_i^\*)^{\otimes 2},
$$
where $\vec{w}^\*\in\Delta^{k-1}$, $\vec{w}^\*\neq \vec{w}$, $\vec{\mu}\_1^\*,\ldots, \vec{\mu}\_k^\* \in \Delta^{d-1}$, and $\vec{\mu}\_1^\*,\ldots, \vec{\mu}\_k^\* \notin \{\vec{\mu}\_1,\ldots, \vec{\mu}\_k\}$. 



Moment matching on the standard simplex
It is a standard result that the matrices of the form $\mu^{\otimes 2}$ for nonzero $\mu$ are the extreme rays of the positive semidefinite cone. That is to say, your condition on the second moments implies that $\mu^{\otimes 2}$ is a scalar multiple of $\mu\_i^{\otimes 2}$ for some $i$. The normalization then gives $\mu = \mu\_i$. So the there never exists a solution to the main question.


