If I define $\mathcal{A} = \{ xx^T : x \in \mathbb{R}^d, \| x \|\_2 \leqslant 1 \}$, then (assuming I recall correctly) it is known that the convex hull of $\mathcal{A}$ is given by


\begin{align}
\text{conv} (\mathcal{A}) = \{ M \in \text{Mat}\_{d \times d} (\mathbf{R}): M = M^T, \| M \|\_\* = 1\},
\end{align}


where $\| \cdot \|\_\*$ is the nuclear norm.


Suppose I now define $\mathcal{A}\_+ = \{ xx^T : x \in \mathbb{R}\_+^d, \| x \|\_2 \leqslant 1\}$, i.e. I restrict to taking outer products of *nonnegative* vectors $x$ with themselves. 


My question is: **Does there exist a similar characterisation of** $\text{conv} (\mathcal{A}\_+)$ **?**




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If context is useful: I'm interested in understanding and characterising when it is possible to write a symmetric probability distributions on two variables as a mixture of i.i.d. distributions, i.e. if $p(x,y) = p(y, x)$, when does there exist 


* some parameter space $\Theta$,
* some family of distributions $\{ p(\cdot | \theta) \}\_{\theta \in \Theta}$, and
* some (prior) distribution $\pi$ such that


\begin{align}
p(x, y) = \int\_{\Theta} p(x|\theta) \, p(y|\theta) \, \pi(\theta) \, d\theta?
\end{align}


I recognise that this has some links to de Finetti's theorem, but I'm not yet sure whether that link can be turned into an answer.



Convex Hull of Outer Products of (Normalised) Nonnegative Vectors
Your characterization of $\text{conv} (\mathcal{A})$ needs one additional restriction---that $M$ is positive semidefinite (the equivalence of these two sets follows fairly quickly from the spectral decomposition).


For $\text{conv} (\mathcal{A}\_+)$, the convex hull is the exact same, but with the positive semidefiniteness requirement replaced by a [complete positivity](http://mathworld.wolfram.com/CompletelyPositiveMatrix.html) requirement. Since checking complete positivity is NP-hard, don't expect an easy way of determining membership in this convex hull.


