Abstract
A compact set c in \({\Bbb R}^d\) is κ-round if for every point \(p\in \partial c\) there exists a closed ball that contains p, is contained in c, and has radius κ diam c. We show that, for any fixed κ > 0, the combinatorial complexity of the union of n κ-round, not necessarily convex, objects in \({\Bbb R}^3\) (resp., in \({\Bbb R}^4\)) of constant description complexity is O(n2+ε) (resp., O(n3+ε)) for any ε > 0, where the constant of proportionality depends on ε, κ, and the algebraic complexity of the objects. The bound is almost tight in the worst case.
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Aronov, B., Efrat, A., Koltun, V. et al. On the Union of κ-Round Objects in Three and Four Dimensions. Discrete Comput Geom 36, 511–526 (2006). https://doi.org/10.1007/s00454-006-1263-x
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DOI: https://doi.org/10.1007/s00454-006-1263-x