More on exposed points and extremal points of convex sets in $\mathbb{R}^n$ and Hilbert space

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