Besicovitch's covering theorem in the parabolic metric

Abstract

We prove that Besicovitch's covering theorem holds for the parabolic metric $d_p((x_1,t_1),(x_2,t_2))=(|x_1-x_2{|}^p+|t_1-t_2{|}^{\frac{p}{2}}{)}^{\frac{1}{p}}$ on ${\mathbb{R}}^n\times \mathbb{R}$ if and only if $p⩾2$. If $p=2$, this answers a question posed by P. Mattila in the affirmative.