The Besicovitch covering theorem for parabolic balls in Euclidean space (Q1730272)

The parabolic metric for $x=(x_1,\ldots,x_n),y=(y_1,\ldots,y_n) \in \mathbb{R}^n$, $n \ge 2$, is defined by $d(x,y)=\max\left\{\sqrt{(x_1-y_1)^2+\ldots+(x_{n-1}-y_{n-1})^2}, \sqrt{|x_n-y_n|}\right\}$. The corresponding parabolic ball with center $x \in \mathbb{R}^n$ and radius $r > 0$ is given by $B(x,r)=\{y \in \mathbb{R}^n: d(x,y) \le r\}$. The following variant of Besicovitch's covering theorem is shown: there is a positive integer $N=N(n)$ such that, for any family $\mathcal{F}$ of parabolic balls in $\mathbb{R}^n$ with uniformly bounded radii, there are $N$ subfamilies $\mathcal{G}_1,\ldots,\mathcal{G}_N \subseteq \mathcal{F}$ such that (i) each $\mathcal{G}_j$ consists of at most countably many disjoint balls and (ii) the set of all centers of balls from $\mathcal{F}$ is covered by $\mathcal{G}_1 \cup \ldots \cup \mathcal{G}_N$.