On the Besicovitch Property for Parabolic Balls

Abstract

Let $ p \ge 1 $ and $ a_1, \dots , a_n $ be positive given numbers. We prove that, the family of all solids of $ {\mathcal R}^n $ of the type $\sum_{i=1}^n \left( \frac{|{x_i}| }{ r^{a_i}} \right)^p < 1 $, $ r > 0 $ satisfies the Besicovitch covering lemma if and only if $ p \ge \frac{\max a_i }{\min a_i } $.