Weak Quadratic Overgroups for Type I Solvable Lie Groups of the Form $\mathbb{R}\ltimes\mathbb{R}^{n}$

Let $G$ be a type I connected and simply connected solvable Lie group defined as the semi-direct product of $\mathbb{R}$ and an $n$-dimensional Abelian ideal $N$ for some $n\geq 1.$ Let $\mathfrak{g}^*/G$ denote the set of coadjoint orbits of $G$, where $\mathfrak{g}^*$ is the dual vector space of the Lie algebra $\mathfrak{g}$ of $G.$ Generally, the closed convex hull of a coadjoint orbit $\mathcal{O}\subset \mathfrak{g}^*$ does not characterize $\mathcal{O}.$ However, we say that a subset $X$ in $\mathfrak{g}^*/G$ is convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in $X.$ In this paper, our main result provides an explicit construction of an overgroup, denoted $G^+,$ containing $G$ as a subgroup and a quadratic map $\varphi$ sending each $G$-orbit in $\mathfrak{g}^*$ to $G^+$-orbit in $(\mathfrak{g}^+)^*,$ in such a manner that the set $\varphi(\mathfrak{g}^*)/G^+$ is convex hull separable, which leads to the separation of elements of $\mathfrak{g}^*/G.$ The Lie group $G^+$ is called a weak quadratic overgroup for $G.$