Lines Avoiding Unit Balls in Three Dimensions

Lines Avoiding Unit Balls in Three Dimensions

Let B be a set of n unit balls in R3. We show that the combinatorial complexity of the space of lines in R3 that avoid all the balls of B is O(n3+ε), for any ε > 0. This result has connections to problems in visibility, ray shooting, motion planning, and geometric optimization.

Unit Distances in Three Dimensions

We show that the number of unit distances determined by n points in R is O(n), slightly improving the bound of Clarkson et al. [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [25].

Lines Tangent to Four Unit Balls problem 2

The Golden Partition Conjecture is motivated by conjectures whose root is a famous sorting problem. Imagine that a finite set S with cardinality n has a hidden total order ≺ that we would like to uncover by making comparisons between pairs of elements from S. Comparing a pair u, v ∈ S means uncovering either u ≺ v or v ≺ u. How many comparisons are needed in the worst case? The classical answer...

On the complexity of the sets of free lines and free line segments among balls in three dimensions

In this paper, we show that the worst-case combinatorial complexity of the set of maximal non-occluded line segments among n unit balls is Θ(n). This improves on the trivial Ω(n) and O(n) bounds and also on the Ω(n) lower bound for the restricted setting of arbitrary-size balls [Devillers and Ramos, 2001]. We also prove an Ω(n) lower bound on the complexity of the set of non-occluded lines amon...

Scaling of Self-Avoiding Walks and Self-Avoiding Trails in Three Dimensions