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The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question:

Given some bounded set $K \subset \mathbb{R}^n$ with $\mathrm{dim}_{\mathcal{H}} K = n-2$, must it be the case that there exists a constant $c = c(n,K) > 0$ for which

$\mathcal{H}^n( (K)_{\rho} ) \geq c\rho^2$

for every $\rho > 0$? (Here $(K)_{\rho} = \bigcup_{x \in K} B_{\rho}^n(x)$.)

If true, it looks like it would be well-known to some... In which case I'm really looking for a reference so as to avoid reproving. Can anyone point me to a reference?

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  • $\begingroup$ Be a little bit careful here: the dimension is too crude a measurement to draw such a conclusion because you may have some extra sub-power decay it does not catch (consider a compact set of dimension $1$ but zero measure on the line and put it in $\mathbb R^3$). However, if the $n-2$-dimensional Hausdorff measure is positive, then it is true and follows straight from the definitions. $\endgroup$
    – fedja
    Commented Feb 13, 2015 at 1:24
  • $\begingroup$ Thanks for the comment fedja, I see your point. I don't know what made me think it was true now. $\endgroup$
    – Spencer
    Commented Feb 13, 2015 at 2:25

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