On Neighbors in Geometric Permutations Shakhar Smorodinsky Tel-Aviv University Joint work with Micha Sharir

Geometric Permutations n A - a set of disjoint convex bodies in R d n A line transversal l of A induces a geometric permutation of A l2l2 l1l l 1 : l 2 :

3 Example 3 12

An example of S with 2n-2 geometric permutations (Katchalski, Lewis, Zaks ) n-2

Motivation? Trust me ……………. There is some!

Problem Statement g d (A) = the number of geometric permutations of A g d (n) = max |A|=n {g d (A)} ? < g d (n) < ?

Known Facts n g 2 (n) = 2n-2 (Edelsbrunner, Sharir 1990) n g d (n) =  (n d-1 ) (Katchalski, Lewis, Liu 1992) n g d (n) = O(n 2d-2 ) (Wenger 1990) n Special cases: n n arbitrary balls in R d have at most  (n d-1 ) GP’s (Smorodinsky, Mitchell, Sharir 1999) n The  (n d-1 ) bound extended to fat objects (Katz, Varadarajan 2001) n n unit balls in R d have at most O(1) GP’s (Zhou, Suri 2001) 4

8 Overview of our result n Define notion of “Neighbors” n Neighbors Lemma: few neighbors => few permutations n In the plane (d = 2) few neighbors n Conjecture: few neighbors in higher dimensions (d > 2)

Definition A- a set of convex bodies in R d A pair (b i, b j ) in A are called neighbors If  geometric permutation for which bi, bj appear consecutive: bibi bjbj

10 Neighbors n-2 Example (2,3) (2,4) ….. (2,n-2)

11 Neighbors No neighbors !!!

Neighbors Lemma In R d, if N is the set of neighbor pairs of A, then g d (A)=O(|N| d-1 ). b1b1 b2b2 h b1b1 b2b2. h Unit Sphere S d-1 b 1 is crossed before b 2 Proof : Fix a neighbor pair ( b 1,b 2 ) in N.

Neighbors Lemma (cont) In R d, if N is the set of neighbor pairs of A, then g d (A)=O(|N| d-1 ). Let P be a set of |N| separating hyperplanes (A hyperplane for each one of the neighbor pairs)

Consider the arrangement of great circles that correspond to hyperplanes in P. A fixed permutation in C C connected component C => Unique GP Unit Sphere S d-1 # connected component < O(|N| d-1 )

Conjecture: The # neighbors of n convex bodies in R d is O(n) If true, implies a  (n d-1 ) upper bound on g d (n)! Else, Return (to SWAT 2004) ;

Upper Bound: #neighbors in the plane O(n) upper bound on the # neighbors in the plane Construct a “neighbors graph”

Neighbors Graph (cont) Connect neighbors as follows: In this drawing rule there may exist crossings

Neighbors Graph (cont) : n However : there are no three pairwise crossing edges (technical proof) Theorem: [Agarwal et al. 1997] A quasi-planar graph with n vertices has O(n) edges. n A graph that can be drawn in the plane with n no three pairwise crossing edges is called a quasi-planar graph.

Further research n Prove: O(n) neighbors in any fixed dimension. n In the plane: Is the neighbors graph planar?