Variational Methods in Image Segmentation pp 94-117 | Cite as

# Tangency Properties of Regular Subsets of ℝ^{N}

## Abstract

In this chapter, we shall be concerned with some tangency properties of regular α-sets of ℝ^{N}. 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).

## Keywords

Space Versus Tangent Space Affine Space Affine Subspace Covering Lemma## Preview

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