  If $C_n(\mathbb{R}^d)$ denotes the configuration space of $n$ distinct points
in $\mathbb{R}^d$, we construct a sequence of maps $(f_m),$ $m \geq 1$, where
\[f_m: C_n(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d\] is real
analytic, and has the property that for any $\mathbf{x} \in C_n(\mathbb{R}^d)$
and any $m \geq 1$, the map $f_m(\mathbf{x},-): \mathbb{R}^d \to \mathbb{R}^d$
is a rational map whose image lies in the convex hull of $\mathbf{x}$. Our
Approximation Conjecture is that for any $\mathbf{x} \in C_n(\mathbb{R}^d)$,
the image of the sphere $S^{d-1}$ under our map $f_m(\mathbf{x},-)$ is an
approximation of the boundary of the convex hull of $\mathbf{x}$. More
precisely, we conjecture that \[ \operatorname{lim}_{m \to \infty}
d_H\left(f_m(\mathbf{x},-)(S^{d-1}), \,\partial \operatorname{Conv}(\mathbf{x})
\right) = 0, \] where $d_H(-,-)$ is the Hausdorff distance,
$\operatorname{Conv}(\mathbf{x})$ is the convex hull of $\mathbf{x}$ and
$\partial$ is the boundary operator. Computer generated plots will be presented
in this work.
