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# Article

 Title: Monotone measures with bad tangential behavior in the plane (English) Author: Černý, Robert Author: Kolář, Jan Author: Rokyta, Mirko Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 52 Issue: 3 Year: 2011 Pages: 317-339 Summary lang: English . Category: math . Summary: 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. (English) Keyword: monotone measure Keyword: monotonicity formula Keyword: tangent measure MSC: 49J45 idZBL: Zbl 1249.49019 idMR: MR2843226 . Date available: 2011-08-15T19:06:39Z Last updated: 2013-10-14 Stable URL: http://hdl.handle.net/10338.dmlcz/141604 . Reference: [1] Černý R.: Local monotonicity of measures supported by graphs of convex functions.Publ. Mat. 48 (2004), 369–380. MR 2091010 Reference: [2] Černý R., Kolář J., Rokyta M.: Concentrated monotone measures with non-unique tangential behaviour in $R^3$.Czechoslovak Math. J.(to appear). MR 2886262 Reference: [3] Kolář J.: Non-regular tangential behaviour of a monotone measure.Bull. London Math. Soc. 38 (2006), 657–666. MR 2250758, 10.1112/S0024609306018637 Reference: [4] Mattila P.: Geometry of Sets and Measures in Euclidean Spaces.Cambridge University Press, Cambridge, 1995. Zbl 0911.28005, MR 1333890 Reference: [5] Preiss D.: Geometry of measures in $\mathbb R^n$: Distribution, rectifiability and densities.Ann. Math. 125 (1987), 537–643. MR 0890162, 10.2307/1971410 Reference: [6] Simon L.: Lectures on geometric measure theory.Proc. C.M.A., Australian National University Vol. 3, 1983. Zbl 0546.49019, MR 0756417 .

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