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Hausdorff dimension of uniformly non-flat sets with topology. (English) Zbl 1065.49028

Let \(d\) be an integer, and let \(E\) be a nonempty closed subset of \(\mathbb{R}^n\). Assume that \(E\) is locally uniformly non-flat, in the sense that for \(x\in E\) and \(r> 0\) small, \(E\cap B(x,r)\) never stays \(\varepsilon_0r\)-close to an affine \(d\)-plane. Also suppose that \(E\) satisfies locally uniformly some appropriate \(d\)-dimensional topological nondegeneracy condition, like Semmes’ condition \(B\). Then the Hausdorff dimension of \(E\) is strictly larger than \(d\). It is seen that this as an application of uniform rectifiability results on Almgren quasiminimal (restricted) sets.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
28A75 Length, area, volume, other geometric measure theory
28A80 Fractals
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