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Investigating the geometric permutations of disjoint convex bodies and the concept of neighbors in Rd, with a focus on unique permutations and bounding the number of permutations. Neighbors Lemma states that few neighbors result in fewer permutations, and the research explores the implications for different dimensions. The study delves into the relationships between neighbor pairs, separating hyperplanes, and connected components to determine the upper bounds and complexities of neighbor configurations. The discussion extends to the construction of a neighbors graph, quasi-planar graphs, and further research questions in different dimensions and graph theories. This research provides valuable insights into the spatial arrangements and connections among convex bodies in geometric permutations.
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On Neighbors in Geometric Permutations Shakhar Smorodinsky Tel-Aviv University Joint work with Micha Sharir
Geometric Permutations • A - a set of disjoint convex bodies in Rd • A line transversal l of A induces a geometric permutation of A l2: <2,3,1> l1: <1,2,3> l2 2 3 1 l1
1 2 3 Example <2,1,3> <2,3,1> <1,2,3>
3 2 n-2 1 An example of S with2n-2 geometric permutations(Katchalski, Lewis, Zaks - 1985) <2,3,…,n-2,1> <3,..2,…,n-2,1>
Motivation? Trust me ……………. There is some!
Problem Statement gd(A) = the number of geometric permutations of A gd(n) = max|A|=n{gd(A)} ? < gd(n) < ?
Known Facts • g2(n) = 2n-2 (Edelsbrunner, Sharir 1990) • gd(n) = (nd-1) (Katchalski, Lewis, Liu 1992) • gd(n) = O(n2d-2) (Wenger 1990) • Special cases: • narbitraryballs in Rd have at most (nd-1)GP’s (Smorodinsky, Mitchell, Sharir 1999) • The (nd-1) boundextended to fat objects (Katz, Varadarajan 2001) • nunit balls in Rd have at mostO(1) GP’s (Zhou, Suri 2001) 4
Overview of our result • Define notion of “Neighbors” • Neighbors Lemma: few neighbors => few permutations • In the plane (d = 2) few neighbors • Conjecture: few neighbors in higher dimensions (d > 2)
bi bj Definition A- a set of convex bodies in Rd A pair (bi, bj) in A are called neighbors If geometric permutation for which bi, bj appear consecutive:
3 2 n-2 1 Neighbors Example (2,3) (2,4) ….. (2,n-2)
Neighbors No neighbors !!!
h b1 b2 Neighbors Lemma In Rd, if Nis the set of neighbor pairs of A, then gd(A)=O(|N|d-1). Proof: Fix a neighbor pair (b1,b2) in N. Unit Sphere Sd-1 h b1 . b2 b1 is crossed before b2
Neighbors Lemma (cont) In Rd, if Nis the set of neighbor pairs of A, then gd(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. connected component C => Unique GP A fixed permutation in C Unit Sphere Sd-1 C # connected component <O(|N|d-1)
The # neighbors of n convex bodies in Rd is O(n) Conjecture: If true, implies a (nd-1)upper bound on gd(n)! Else, Return (to SWAT 2004);
1 5 4 2 3 O(n) upper bound on the # neighbors inthe plane Upper Bound: #neighbors in the plane Construct a “neighbors graph”
1 5 4 2 3 Connect neighbors as follows: Neighbors Graph (cont) Inthis drawing rule there may existcrossings
Neighbors Graph (cont) : • However: there are no three pairwise crossing edges (technical proof) A graph that can be drawn in the plane with no three pairwise crossing edges is called a quasi-planar graph. Theorem: [Agarwal et al. 1997] A quasi-planar graph with nvertices has O(n) edges.
Further research • Prove: O(n) neighbors in any fixed dimension. • In the plane: Is the neighbors graphplanar?
