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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...

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 ${ℳ}_p^{\alpha }\left(X\right):={\int }_X{\int }_X{\int }_X{\kappa }^p(x,y,z)d_X^{\alpha }\left(x\right)d_X^{\alpha }\left(y\right)d_X^{\alpha }\left(z\right)$, where κ(x,y,z) is the inverse circumradius of the triangle defined by x,y and z, we find that ${ℳ}_p^{\alpha }\left(X\right)<\infty$ 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...