Geometric Measure Theory On Euclidean Spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T01:40:41Z http://mathoverflow.net/feeds/question/102887 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102887/geometric-measure-theory-on-euclidean-spaces Geometric Measure Theory On Euclidean Spaces Jeremy Young 2012-07-22T18:11:02Z 2012-07-23T07:51:27Z <p>Dear all,</p> <p>I have recently found the following discussion: <a href="http://mathoverflow.net/questions/102061/hausdorff-measure-and-minkowskis-content-boundary-measures" rel="nofollow">http://mathoverflow.net/questions/102061/hausdorff-measure-and-minkowskis-content-boundary-measures</a> regarding different boundary measures on $\mathbb{R}^n$ . </p> <p>The discussion made me wondering:</p> <p>Is there any example in the opposite direction?</p> <p>i.e. an example of a set $A \subseteq \mathbb{R} ^n$ such that $Leb^+ (A) > H^{n-1} (\partial A)$ ? </p> <p>Thanks in advance ! </p> http://mathoverflow.net/questions/102887/geometric-measure-theory-on-euclidean-spaces/102891#102891 Answer by Tapio Rajala for Geometric Measure Theory On Euclidean Spaces Tapio Rajala 2012-07-22T19:22:04Z 2012-07-22T19:22:04Z <p>Take $[0,1]\times \{0\} \subset \mathbb{R}^2$.</p> http://mathoverflow.net/questions/102887/geometric-measure-theory-on-euclidean-spaces/102892#102892 Answer by Anton Petrunin for Geometric Measure Theory On Euclidean Spaces Anton Petrunin 2012-07-22T19:23:20Z 2012-07-23T07:51:27Z <p>$A=\mathbb{Z}\subset \mathbb R^2$. In this case $\partial\mathbb{Z}= \mathbb{Z}$ and $\mathop{Leb}^+(\mathbb Z)=\infty$ and $H^1(\mathbb Z)=0$.</p> <p>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 $\sqrt[3]{\varepsilon}$.</p>