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** $\text{conv} (\mathcal{A}_+)$ **?**

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.