**
Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 1141-1167, 2011
**

#
Concentrated monotone measures with

non-unique tangential behavior in $\Bbb R^3$

##
Robert Černý, Jan Kolář, Mirko Rokyta

* R. Černý*, Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: ` rcerny@karlin.mff.cuni.cz`; * J. Kolář*, Institute of Mathematics, Czech Academy of Sciences, ®itná 25, 115 67 Praha 1, Czech Republic, e-mail: ` kolar@math.cas.cz`; * M. Rokyta*, Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: ` rokyta@karlin.mff.cuni.cz`

**Abstract:** We show that for every $\varepsilon>0$ there is a set $A\subset\Bbb R^3$ such that ${\Cal H}^1\llcorner A$ is a monotone measure, the corresponding tangent measures at the origin are non-conical and non-unique and ${\Cal H}^1\llcorner A$ has the $1$-dimensional density between $1$ and $2+\varepsilon$ everywhere in the support.

**Keywords:** monotone measure, monotonicity formula, tangent measure

**Classification (MSC 2010):** 53A10, 49Q15, 28A75

**Full text** available as PDF.

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