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 $(f_m),$ $m\ge 1$, 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 \[ \operatorname{lim}_{m \to \infty} d_H\left(f_m(\mathbf{x},-)(S^{d-1}), \,\partial \operatorname{Conv}(\mathbf{x}) \right) = 0, \] where $d_H(-,-)$ is the Hausdorff distance, $\mathrm{Conv}(\mathbf{x})$ is the convex hull of $\mathbf{x}$ and $\partial$ is the boundary operator. Computer generated plots will be presented in this work.

The paper is withdrawn: the conjecture here is false, but it led to another work with Peter J. Olver, arxiv:2007.03011 [math.MG], for which the corresponding statement is actually proved there