Let $f=(f_0,\ldots f_n)$ be a vector in $\Bbb N^{n+1}$. Let $X$ be the set of all (ordered) $f_0$-tuples in $\Bbb R^n$ whose convex hull has $f$ as its $f$-vector. Assume that $X$ is non-empty. Is anything known about $X$? For instance, is it algebraic? Semi-algebraic?

Let $f=(f_0,\ldots f_n)$ be a vector in $\Bbb N^{n+1}$. Let $X$ be the set of all (ordered) $f_0$-tuples in $\Bbb R^n$ whose convex hull has $f$ as its $f$-vector. Assume that $X$ is non-empty. Is anything known about $X$? For instance, is it algebraic? Semi-algebraic?

Edit: By “is $X$ algebraic”, I mean: Is $X$ locally closed, or maybe just constructible, in the Zariski topology on $\Bbb R^{f_0 n}$?