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Results tagged with hausdorff-dimension
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user 25510
Questions about dimensions of possibly highly irregular or "rough" sets, Hausdorff–Besicovitch dimension and related concepts such as box-counting or Minkowski–Bouligand dimension.
6
votes
Accepted
Hausdorff dimension of convex set in ${\bf R}^n$
Yes. The boundary even has a locally finite Haudsorff $(m-1)$-measure.
No. A convex function of 1 variable has increasing derivative, but this derivative can have a
dense set of jumps.
In general, …
- 91.8k
5
votes
Plane curve with continuously increasing Hausdorff dimension
I do not have a reference but can propose a proof of Proposition 1 which is independent of and simpler than Proposition 2.
Let us first construct a non-closed curve, namely a graph of a convex functi …
- 91.8k
3
votes
Hausdorff dimension of Julia sets of quadratics not in the Mandelbrot set.
At some points of the boundary of M, the Hausdorff dimension is not continuous, these are points
of "parabolic implosion", see MR2521938.
- 91.8k
2
votes
Hausdorff measure of the zero set
Yes, it is true. If the zero set has Hausdorff dimension $<n-1$ then almost every line in the direction of a coordinate axis will not intersect the set. This easily follows from the definition of the …
- 91.8k
0
votes
Fractal questions: Weierstraß-Mandelbrot
The word "fractal" has no established commonly accepted definition. (Some definitions involve self-similarity, others only Hausdorff dimension).
You should specify what exactly you mean by Weierstra …
- 91.8k