Let the set $\mathcal{S}=\left\{ \sum_{i=1}^n x_i\mathbf{h}_i：x_i\in[0,1] \text{ for all }i\right\}\subset\mathbb{R}^r$, i.e., a zonotope generated by $n$ column vector $\mathbf{h_1},\cdots,\mathbf{h}_n$ in Euclidean $r$-space. For simplicity, we denote $\mathsf{H}=[\mathbf{h_1},\cdots,\mathbf{h}_n]$ of rank $r$. Let $P$ is a probability measure on the support set $\mathcal{S}$.

For any realization $\mathbf{s}\in \mathcal{S}$, we let $\mathcal{X}_{\mathbf{s}}=\left\{ \mathbf{x}\in [0,1]^n: \mathsf{H}\mathbf{x}=\mathbf{s} \right\}$, which must be nonempty and convex. $\phi$ is a mapping from $\mathcal{S}$ onto the unit cube satisfying $\phi(\mathbf{s})\in \mathcal{X}_{\mathbf{s}}$ for all $\mathbf{s}\in \mathcal{S}$. Under a given probability measure, the vector-valued expectation of $\phi(\mathbf{s})$ is \begin{equation} \mathbf{f}_{P}(\phi)=\int \phi(\mathbf{s})\,dP. \end{equation} Over all admissible mappings $\phi$, we can construct a set $\mathcal{E}_P=\left\{ \mathbf{f}_{P}(\phi)\right\} \subset [0,1]^n$. The convexity of $\mathcal{E}_P$ can be easily verified.

My question is given a point $\mathbf{a}\in [0,1]^n$ how to judge whether $\mathbf{a}$ lies in $\mathcal{E}_P$. Furthermore, I want to know a sufficient and necessary condition of the probability measure $P$ such that $\mathbf{a}\in \mathcal{E}_P$. Some geometric and topological properties of $\mathcal{E}_P$ are welcome.

Personal remark:

Since the matrix $ \mathsf{H}$ has full row rank, the MP inverse of $ \mathsf{H}$ is $\mathsf{H}^+=\mathsf{H}^T(\mathsf{H}\mathsf{H}^T)^{-1}$. The set $\mathcal{X}_{\mathbf{s}}$ actually is the intersection of the affine set $\text{null}(\mathsf{H})+\mathsf{H}^+\mathbf{s}$ and a unit $n$-cube.

The case is trivial when $\mathsf{H}$ is invertible.