Prove that following two sets are equal: $$ \operatorname{conv}\left\{\, xx^T \,\middle|\, x\in\Bbb R^n, \|x\|=1 \,\right\} = \left\{ A \in S_n^+ \,\middle|\, \operatorname{Tr}(A)=1 \,\right\}, $$ where $S_n^+$ is the set of all positive-semidefinite matrices.
I already proved that second one contains the first (it is quite easy), but I have no idea how to prove that every positive-semidefinite matrix is the weighted sum of outer products.
Every symmetric matrix can be decomposed as follows: $$ {\displaystyle A=Q\Lambda Q^{\textsf {T}}}, $$
where $\Lambda$ is diagonal
We can construct a diagonal matrix from outer products easily, taking vectors with only one non-zero entry of 1 and coefs as diagonal entries of the matrix. Then we just insert $Q$ and $Q^{\textsf {T}}$ into the sum and get the weighted sum of outer products.