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Tangency properties of sets with finite geometric curvature energies

100%

Scholtes S.

Fundamenta Mathematicae

2012

tom 218

nr 2

165-191

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.

Discrete thickness

100%

Scholtes S.

Molecular Based Mathematical Biology

2014

tom 2

nr 1

We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are deffned on equilateral polygons with n vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the Γ-limit of the discrete ropelength for n → ∞, regarding the topology induced by the Sobolev norm ‖ · ‖ W1,∞(S1,ℝd). This result directly implies the convergence of almost minimizers of the discrete energies in a fixed knot class to minimizers of the smooth energy.Moreover,we show that the unique absolute minimizer of inverse discrete thickness is the regular n-gon.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Molecular Based Mathematical Biology

2 Scholtes S.

1 2014

1 2012

Wyniki wyszukiwania

Tangency properties of sets with finite geometric curvature energies

Discrete thickness