Abstract
Assume that lower and upper d-dimensional Minkowski contents of A⊂ R N are different both from 0 and ∞. We show that the function d( x, A) − γ is integrable in a tubular neighbourhood of A if and only if γ< N− d (the if part is known). von Koch’s curve and the Sierpinski gasket are shown to satisfy the Minkowski content condition. The notions of relative Minkowski content and relative box dimension are introduced in order to extend this result to singular functions generated by more general fractal sets, which may have classical upper or lower d-dimensional Minkowski content equal to 0 or ∞.
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