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