Geodesic
Monotone measures with bad tangential behavior in the plane
Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 3, pp. 317-339 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We show that for every $\varepsilon > 0$, there is a set $A\subset \mathbb R^2$ such that $\mathcal H^1 \llcorner A$ is a monotone measure, the corresponding tangent measures at the origin are not unique and $\mathcal H^1 \llcorner A$ has the $1$-dimensional density between $1$ and $3+\varepsilon $ everywhere on the support.
We show that for every $\varepsilon > 0$, there is a set $A\subset \mathbb R^2$ such that $\mathcal H^1 \llcorner A$ is a monotone measure, the corresponding tangent measures at the origin are not unique and $\mathcal H^1 \llcorner A$ has the $1$-dimensional density between $1$ and $3+\varepsilon $ everywhere on the support.
Classification : 49J45
Keywords: monotone measure; monotonicity formula; tangent measure 
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     title = {Monotone measures with bad tangential behavior in the plane},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {317--339},
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Černý, Robert; Kolář, Jan; Rokyta, Mirko. Monotone measures with bad tangential behavior in the plane. Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 3, pp. 317-339. http://geodesic.mathdoc.fr/item/CMUC_2011_52_3_a0/