Tubular Neighborhoods and Minkowski Measurability

Abstract

In this chapter, we obtain (in Section 8.1) a distributional formula for the volume of the tubular neighborhoods of the boundary of a fractal string, called a tube formula. In Section 8.1.1, under more restrictive assumptions, we also derive a tube formula that holds pointwise. In Section 8.3, we then deduce from these formulas a new criterion for the Minkowski measurability of a fractal string, in terms of its complex dimensions. Namely, under suitable assumptions, we show that a fractal string is Minkowski measurable if and only if it does not have any nonreal complex dimensions of real part D, its Minkowski dimension. This completes and extends the earlier criterion obtained in [LapPo1-2].