maximal epsilon packing is also a net

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Lemma

If $X$ is an epsilon packing that is maximal, then $X$ is a $2 ϵ$ net.

Maximal

We say that $X$ is maximal if $X \cup {x^{'}}$ is not a packing for all $x^{'} \notin X$

Proof

Let $y \in Y ∖ X$ . Maximality means $B^{d} (y, ϵ) \cap B^{d} (x, ϵ) \neq \emptyset$ for some $x \in X$ . Let $z$ be in this intersection

Then by the triangle inequality and by construction of our balls above, we get

| | y - x | | \leq | | y - z | | + | | z - x | | \leq 2 ϵ

\begin{matrix} ◼ \end{matrix}

File	Last Modified
Random Matrix Lecture 04	2025-09-13
we can find an epsilon net with the bound for an epsilon packing	2025-09-04

maximal epsilon packing is also a net

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