 
Summary: On the Union of #Round Objects
in Three and Four Dimensions #
Boris Aronov + Alon Efrat # Vladlen Koltun § Micha Sharir ¶
April 1, 2004
Abstract
A compact body c in R d is #round if for every point p # #c there exists a closed
ball that contains p, is contained in c, and has radius # diam c. We show that, for any
fixed # > 0, the combinatorial complexity of the union of n #round, not necessarily
convex objects in R 3 (resp., in R 4 ) of constant description complexity is O(n 2+# ) (resp.,
O(n 3+# )) for any # > 0, where the constant of proportionality depends on #, #, and
the algebraic complexity of the objects. The bound is almost tight.
1 Introduction
Given a set C of n geometric objects in R d , let U = U(C) :=
# c#C c denote their union, and
let A = A(C) denote the arrangement [39] of the (boundaries of the) objects in C. The
(combinatorial) complexity of U is defined to be the number of faces of A of all dimensions
on the boundary #U of the union. The study of the complexity of the union of objects in
two dimensions has a long and rich history in computational and combinatorial geometry,
starting with the results of Kedem et al. [29] and Edelsbrunner et al. [19], who have shown
# Work on this paper by B.A. and M.S. has been supported by a grant from the U.S.Israeli Binational
