# Besicovitch covering theorem in a generic finite dimensional normed vector space

Let $$\left(V,\|\cdot\|\right)$$ be a finite dimensional normed vector space. If $$A \subset V$$ denote by $$\chi_A$$ the indicator function of the set $$A$$ and if $$r>0$$ and $$v\in V$$ denote by $$\bar{B}_r(v)$$ the closed ball of $$\left(V,\|\cdot\|\right)$$ centered in $$v$$ of radius $$r$$.

Is it true that there exists $$N\in\mathbb{N}$$ such that for each bounded $$E \subset V$$ and each $$r\colon E \to (0,+\infty)$$, there exists a countable subset $$Q$$ of $$E$$ such that $$\chi_E\subset\sum_{v\in Q} \chi_{\bar{B}_{r(v)}(v)}\le N?$$

I know that the result, known as Besicovitch covering theorem, holds true both in $$(\mathbb{R}^n,\|\cdot\|_2)$$ and in $$(\mathbb{R}^n,\|\cdot\|_\infty)$$, where $$\|\cdot\|_2$$ is the euclidean norm and $$\|\cdot\|_\infty$$ is the sup norm. Can anyone provide any reference for the general case?

• There are some papers by Rigot-Preiss-Tiser and Le Donne-Rigot about Besicovitch covering theorems in Banach spaces and the Heisenberg group respectively. I am pretty sure that they adress there the state of the art in f.d. normed spaces. Commented Feb 25, 2020 at 10:19