Sharp Bounds on Geometric Permutations for Pairwise Disjoint Balls in R d Shakhar Smorodinsky Joe Mitchell Micha Sharir

Geometric Permutations nAnA - a finite set of pairwise disjoint convex bodies in R d nAnA line transversal l of A induces a geometric permutation of A nlnl 1 : <1,2,3> = <3,2,1> l 2 : <2,3,1> = <1,3,2> l2l2 l1l A

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

Known Facts ngng 2 (n) = 2n-2 (Edelsbrunner, Sharir 1990) ngng d (n) = (n d-1 ) (Katchalski, Lewis, Liu 1992) ngng d (n) = O(n 2d-2 ) (Wenger 1990) 4

An example of S with g 2 (S) = 2n-2 (Katchalski, Lewis, Zaks ) S 5

What have we done? nSnSharp bounds on the number of geometric permutations for n pairwise disjoint balls in R d ; (n d-1 ). 6

The Technique n Separation set 7 A

Definition. nAnA- a set of convex bodies in R d nPnP - a set of hyperplanes in R d P separates A if for every pair of bodies b i,b j in A, there is a hyperplane h, in P such that h separates b i and b j h bibi bjbj 8

Useful Lemma nInIn R d, if P separates A then g d (A)=O(|P| d-1 ). nPnProof: Fix two bodies b 1,b 2 in A. b1b1 b2b2 h b1b1 b2b2. h Unit Sphere S d-1 B 1 is crossed before B 2

Consider the arrangement of great circles that correspond to hyperplanes in P. n A connected component C, corresponds to a set of line orientations with at most one geometric permutation. A fixed permutation in C C 10

Separating Disjoint Balls n S - set of n pairwise disjoint balls in R d n There exists a separation set for S of size O(n). n Constant in the O notation depends on the dimension d. 11

Note: nInIn the general case of n pairwise disjoint convex bodies in R d (d>2) there are families which cannot be separated with less than (n 2 ) hyperplanes. 12

bibi bjbj Constructing a separation set for balls nSnS={B 1,...,B n } nrnr 1 >r 2 >...>r n nCnC={C 1,...,C K } a covering of S d-1 by a set of K spherical caps of small diameter h i,k C k (b i )

14 Proof: Induction on the size of B i b i+1 bjbj j,j’i CkCk b j’ h h’  ’’

Lower Bounds

Further research nSnSeparating “fat” convex bodies with linear number of hyperplanes. nSnSeparating “neighbors” only. nCnConstant number of permutations for unit balls in R d n (n (For d=2 and n large enough, there are at most two geometric permutations (Smorodinsky, Sharir, Mitchell; Asinowsky, Katchalski 1998). 18