Holes in planar parallel sets: An integrated Betti-number bound and its pointwise failure

Abstract

Let A be a nonempty compact subset of the plane and let A (r) be its parallel set at distance r. We prove that the number of holes of A (r) the number of bounded components of its complement, which is its rst Betti number satises $\infty$ r 0 $β$1(A (r) ) dr $\le$ 4050 (diam A) 4 r -3 0 for every r0 > 0, the integrand vanishing for r $\ge$ diam A/ $\sqrt$ 3. The proof rests on Fu's theorem that the critical values of the distance function of a planar compact set form a set of vanishing half-dimensional Hausdor measure, on two lemmas of Rataj, Spodarev and Meschenmoser, and on a square-root summability estimate for the gaps of the critical-value set, of which we give a complete proof. We show by an explicit family of curves two combs facing each other that no analogous bound can hold at a xed radius: a connected curve of bounded length, diameter, oscillation count, parallel-set area and parallel-set perimeter can have arbitrarily many holes at one radius, so the integrated estimate cannot be replaced by a xed-radius bound in terms of these coarse geometric quantities. We also bound the hole count uniformly in the radius by the number of components of local maxima of the distance function, and record a bound on the boundary length of a parallel set by its area. The results supply the deterministic input for limit theorems on the persistent homology of the Wiener sausage.