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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
##### Keywords:
rectifiable set; Jones numbers; Hausdorff measure; Lipschitz map
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##### References:
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