Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
monotone measure; monotonicity formula; tangent measure
monotone measure; monotonicity formula; tangent measure
Summary:
We show that for every $\varepsilon >0$ there is a set $A\subset \mathbb{R}^3$ 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.
References:
[1] Černý, R.: Local monotonicity of Hausdorff measures restricted to curves in $\mathbb R^n$. Commentat. Math. Univ. Carol. 50 (2009), 89-101. MR 2562806
[2] Černý, R., Kolář, J., Rokyta, M.: Monotone measures with bad tangential behavior in the plane. Commentat. Math. Univ. Carol. 52 (2011), 317-339. MR 2843226
[3] Kolář, J.: Non-regular tangential behaviour of a monotone measure. Bull. London Math. Soc. 38 (2006), 657-666. DOI 10.1112/S0024609306018637 | MR 2250758
[4] Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Camridge Studies in Advanced Mathematics 44. Cambridge University Press Cambridge (1995). MR 1333890
[5] Preiss, D.: Geometry of measures in $\mathbb R^n$: Distribution, rectifiability and densities. Ann. Math. 125 (1987), 537-643. DOI 10.2307/1971410 | MR 0890162
[6] Simon, L.: Lectures on Geometric Measure Theory. Proc. C. M. A., Vol. 3. Australian National University Canberra (1983). MR 0756417
Search
Partner of
