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2012 | 218 | 2 | 165-191
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Tangency properties of sets with finite geometric curvature energies

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We investigate tangential regularity properties of sets of fractal dimension, whose inverse thickness or integral Menger curvature energies are bounded. For the most prominent of these energies, the integral Menger curvature
$ℳ_{p}^{α}(X): = ∫_{X}∫_{X}∫_{X} κ^{p}(x,y,z) d𝓗 ^{α}_{X}(x)d𝓗 ^{α}_{X}(y)d𝓗 ^{α}_{X}(z)$,
where κ(x,y,z) is the inverse circumradius of the triangle defined by x,y and z, we find that $ℳ_{p}^{α}(X) < ∞$ for p ≥ 3α implies the existence of a weak approximate α-tangent at every point of the set, if some mild density properties hold. This includes the scale invariant case p = 3 for $ℳ ¹_{p}$, for which, to the best of our knowledge, no regularity properties have been established before. Furthermore we prove that for α = 1 these exponents are sharp, i.e., if p lies below the threshold value of scale invariance, then there exists a set containing points with no weak approximate 1-tangent, but such that the energy is still finite. Moreover we demonstrate that weak approximate tangents are the most we can expect. For the other curvature energies analogous results are shown.
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  • Institut für Mathematik, RWTH Aachen University, Templergraben 55, D-52062 Aachen, Germany
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