Rational Maps and Boundaries of Convex Hulls

Abstract

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.