# How to judge whether the following convex set contains a given point?

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:

1. 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.

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