Rational Maps and Boundaries of Convex Hulls

If $C_n({\mathbb{R}}^d)$ denotes the configuration space of $n$ distinct points in ${\mathbb{R}}^d$, we construct a sequence of maps $(f_m),$ $m\ge 1$, 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 $m\ge 1$, 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 $${\lim }_{m\to \mathrm{\infty }}{d}_H(f_m(\mathbf{x},-)(S^{d-1}),\mathrm{\partial }\mathrm{Conv}(\mathbf{x}))=0,$$ where $d_H(-,-)$ is the Hausdorff distance, $\mathrm{Conv}(\mathbf{x})$ is the convex hull of $\mathbf{x}$ and $\mathrm{\partial }$ is the boundary operator. Computer generated plots will be presented in this work.