Tangency Properties of Regular Subsets of ℝN
In this chapter, we shall be concerned with some tangency properties of regular α-sets of ℝN. We shall prove that a is necessarily an integer and that a regular set admits a tangent affine space almost everywhere and therefore “looks like” an α-dimensional manifold almost everywhere. The proof is essentially based on a remarkable reflection lemma due to Marstrand. Roughly speaking, this lemma, which is not difficult to prove once the right ideas are at hand, says that given two points x and y in a regular set A, the reflected point 2x-y lies close to A. This argument can be iterated by successive reflections and therefore yields nets of points which are contained in an affine sub space V and close to A. Of course, the whole argument is a little bit more technical than this abstract. We shall need precise estimates on how a net of points obtained by reflection spreads out and on how close to A the reflected point 2x-y is. This will be done in the first section. The second section is devoted to the existence proof, for any r > 0, of “approximate tangent affine spaces”, V(x,r), at most points x ∈ A. The third section analyses accurately how close A is to its approximate tangent spaces. It is proved that the orthogonal projection of A onto V(x, r) has a density inside B(x,r) tending to 1. This fact easily implies convergence of the sets V(x, r) towards an (almost everywhere) unique tangent space V(x).
KeywordsSpace Versus Tangent Space Affine Space Affine Subspace Covering Lemma
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