 
Summary: On the Union of #Round Objects
in Three and Four Dimensions #
Boris Aronov + Alon Efrat # Vladlen Koltun § Micha Sharir ¶
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.
Categories and Subject Descriptors: F.2.2 [Theory of
Computation]: Nonnumerical Algorithms and Problems
geometrical problems and computations; G.2.1 [Discrete Math
ematics]: Combinatoricscombinatorial complexity
General Terms: Theory, algorithms
Keywords: Combinatorial complexity, union of objects, fat
objects
