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Quantitative conditions of rectifiability. (Conditions quantitatives de rectifiabilité.) (French) Zbl 0890.28004
A set \(E\subset {\mathbf R}^n\) is said to be \(d\)-rectifiable if \(E\subset E_{0}\cup \bigcup \Gamma_{i}\), with the \(d\)-dimensional Hausdorff measure of \(E_{0}\) equal to zero and \(\Gamma_{i}=\{x+A_{i}(x);x\in P_{i}\}\), where \(A_i:P_i\rightarrow P_i^\perp\) is a Lipschitz map and \(P_{i}\) is a \(d\)-dimensional subspace of \({\mathbf R}^n\). Define the numbers \(\beta_{\infty}\), of P. W. Jones, as \[ \beta_{\infty} (x,t,E)=\inf_{P} \sup_{y\in E\cap B(x,t)} \biggl( {\text{ dist}(y,P) \over t}\biggr), \] if \(E\cap B(x,t)\not= \emptyset\); if the intersection is empty, then \(\beta_{\infty} (x,t,E)=0.\) For \(1\leq q < \infty\), \(\beta_{q}\) is defined as \[ \beta_{q}(x,t,E)=\inf_{P} \Biggl({1 \over t^{d}}\int_{E\cap B(x,t)} \biggl({\text{ dist}(y,P) \over t}\biggr)^{q} dH^{d}(y) \Biggr)^{1/q}. \] The infimum is taken over all \(d\)-planes in \({\mathbf R}^{n}\). The number \(q\) is to be chosen as follows: if \(d=1\) then \(1\leq q \leq \infty\), if \(d>1\) then \(1\leq q < 2d/(d-2)\).
The main result is the following:
Let \(E\subset {\mathbf R}^n\) be a compact set with \(H^d(E)<\infty\) and suppose that at \(H^{d}\)-almost all \(x\in E\), \[ \Theta_{*}^{d}(x,E)=\liminf_{r\downarrow 0} {H^{d}(E\cup B(x,r) )\over (2r)^{d}}>0, \] \[ \int^{1}_{0}\beta_{q}(x,t,E)^{2}{\text{ d}t \over t}<\infty. \] Then the set \(E\) is \(d\)-rectifiable.
Reviewer: O.Svensson (Lulea)

MSC:
28A75 Length, area, volume, other geometric measure theory
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