Geometric Measure Theory On Euclidean Spaces - MathOverflow

Geometric Measure Theory On Euclidean Spaces

2012-07-22T18:11:02Z

Dear all,

I have recently found the following discussion: http://mathoverflow.net/questions/102061/hausdorff-measure-and-minkowskis-content-boundary-measures regarding different boundary measures on $\mathbb{R}^n $ .

The discussion made me wondering:

Is there any example in the opposite direction?

i.e. an example of a set $A \subseteq \mathbb{R} ^n $ such that $ Leb^+ (A) > H^{n-1} (\partial A) $ ?

Thanks in advance !

Answer by Tapio Rajala for Geometric Measure Theory On Euclidean Spaces

2012-07-22T19:22:04Z

Take $[0,1]\times \{0\} \subset \mathbb{R}^2$.

Answer by Anton Petrunin for Geometric Measure Theory On Euclidean Spaces

2012-07-22T19:23:20Z

$A=\mathbb{Z}\subset \mathbb R^2$. In this case $\partial\mathbb{Z}= \mathbb{Z}$ and $\mathop{Leb}^+(\mathbb Z)=\infty$ and $H^1(\mathbb Z)=0$.

You can get a bounded example of the same type. Take a countable nowhere dense set $A$ in the unit disc such that the $\varepsilon$-neighborhood of $A$ contains a disc of radius $\sqrt[3]{\varepsilon}$.