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Let $𝕍$ be a separable real Hilbert space, $k\in \mathbb{N}$ with $k<dim𝕍$, and let $B$ be convex and closed in $𝕍$. Let $𝒫$ be a collection of linear $k$-subspaces of $𝕍$. A point $w\in B$ is called exposed by $𝒫$ if there is a $P\in 𝒫$ so that $(w+P)\cap B=\left\{w\right\}$. We show that, under some natural conditions, $B$ can be reconstituted as the convex hull of the closure of all its exposed by $𝒫$ points whenever $𝒫$ is dense and $G_{\delta }$. In addition, we discuss the question when the set of exposed by some $𝒫$ points forms a $G_{\delta }$-set.