the desired upper bound on $\mathfrak{W}$.

Finally, $m^d \leq s^d /r^d$, giving the uniform upper bound on $|N_I|$. Since $\tilde{\tau}_s \leq |N_I|$, the final claim of the lemma follows.

As a simple corollary to the lemma, we see that the hypothesis of Theorem \[thm1\] implies the hypothesis of Theorem \[thm2\], as long as $\tilde{\delta} < \delta$ and $s$ is sufficiently large:

\[DeltaCor\] For any $\tilde{\delta} <\delta$, there exists $s_0$ depending only on $\delta$, the chosen norm, and the dimension, such that whenever $s\ge s_0$, $|E| >
(1+\delta) \mu$ implies $|E_s| > (1+\tilde{\delta})\tilde{\mu}_s$.

Pick $s$ large enough to ensure $(1 +\tilde{\delta}) (1 + C/s) \leq 1 + \delta$. Lemma \[lmm3\] guarantees that $|E_s| \geq |E|$ and $$|E| \geq (1+\delta) \mu \geq (1 +\tilde{\delta}) (1 + C/s) \mu \geq (1 +\tilde{\delta}) \tilde{\mu}_s$$ This proves the corollary.

Next, we get some quantitative bounds on the geometric approximation of certain subsets of $[0,1]^d$ by the $A_I$’s. Let $S$ be a convex subset of a ball of diameter $r$. Assume that $$z(S) := \frac{\lambda(S)}{\tau} \geq \varepsilon/16\, .$$ Recall that $\tau$ is the volume of a ball of diameter $r$. We rescale Lebesgue measure to bypass factors of $\tau$ throughout this section. The first goal is 