Given an $n$-dimensional vector $\mathbf{c}\in [0,1]^n$, let $\Delta_{\mathbf{c}}$ be the set of points $\{\mathbf{x}\in [0,1]^n: \langle \mathbf{c},\mathbf{x} \rangle \le 1\}$, where $\langle \mathbf{c},\mathbf{x} \rangle$ is the inner product between $\mathbf{c}$ and $\mathbf{x}$.

**Question**: Given $\mathbf{c}$ and a vector $\mathbf{z}\in [0,1]^n$, how can we *efficiently* compute the projection $P(\mathbf{z}, \Delta_{\mathbf{c}})$ of $\mathbf{z}$ onto $\Delta_{\mathbf{c}}$?

By writing $P(\mathbf{z}, \Delta_{\mathbf{c}})$, we mean $\arg\min_{\mathbf{z'}\in\Delta_{\mathbf{c}}} \Vert \mathbf{z'}-\mathbf{z} \Vert$, where $\Vert \cdot \Vert$ denotes the regular Euclidean norm.

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