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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?

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  • $\begingroup$ 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. $\endgroup$
    – Diesirae92
    Feb 25, 2020 at 10:19

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The result is claimed to hold true (as a consequence of the theory developed in Federer - Geometric Measure Theory about directionally limited sets) in Heinonen - Lectures on analysis on metric spaces, theorem 1.14 and example 1.15

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