Line Transversals to Disjoint Balls (Joint with C. Borcea, O. Cheong, X. Goaoc, A. Holmsen) Sylvain Petitjean LORIA, Vegas team O. Schwarzkopf

A long time ago…  E. Helly (1923): n convex sets in R d have a point in common iff every d+1 have a point in common.  Basic combinatorial result on convex sets  E. Helly (1923): n convex sets in R d have a point in common iff every d+1 have a point in common.  Basic combinatorial result on convex sets

Transversals  Reformulate: points hitting convex sets  Raises the obvious question: can one generalize to lines hitting convex sets? (line transversals)  No! Bummer.  Raises the obvious question: can one generalize to lines hitting convex sets? (line transversals)  No! Bummer.

Milestones  Danzer (1957): n disjoint unit discs in R 2 have a line transversal if and only if every 5 discs have a line transversal.  shape is important, convexity not enough  Danzer (1957): n disjoint unit discs in R 2 have a line transversal if and only if every 5 discs have a line transversal.  shape is important, convexity not enough  Hadwiger (1957): n disjoint convex sets in R 2 have a line transversal if and only if every triple has a transversal consistent with some fixed order  order is important  Hadwiger (1957): n disjoint convex sets in R 2 have a line transversal if and only if every triple has a transversal consistent with some fixed order  order is important

In 3D: bummer again!  Holmsen-Matousek (2004): No Helly-type theorem for translates of convex sets, not even with a restriction on the ordering (à la Hadwiger) geometric permutations ≠ isotopy equiv. induced by ordering ≠ equiv. induced by connected components

What about balls?  Danzer’s conjecture: Helly for disjoint balls in nD typeHadwiger numberHelly number Hadwiger (1957) & Grünbaum (1960) thinly distributed in R d d 2  2d-1 Holmsen et al. (2003) & Cheong et al. (2005) disjoint unit in R 3 12  646  11 Cheong et al. (2006)pairwise-inflatable in R d 2d4d-1 Borcea et al. (2007)disjoint in R d 2d 

Convexity of cone of directions  Borcea, Goaoc, P. (2007): Directions of oriented lines stabbing a finite family of disjoint balls in R d in a given order form a strictly convex subset of S d-1  Instrumental in most proofs in transversal theory  Previously known for thinly distributed balls (Hadwiger), pairwise-inflatable balls

3D case: 3 disjoint balls

New proof technique  Write down equations  conics and sextic  Write down equations  conics and sextic  Identify the border arcs  Prove Hessian does not meet them  local convexity  Prove Hessian does not meet them  local convexity  Argue that cone is contractible

Disjointness is a natural boundary

Extension to higher dimensions

Implications: disjoint balls  Isotopy = geometric permutations  Smorodinsky et al. (2000): n disjoint balls in R d have  (n d-1 ) geometric permutations in the worst case  same bound for connected components, previous was O(n 2d-4 ); also better bound in R 3  Isotopy = geometric permutations  Smorodinsky et al. (2000): n disjoint balls in R d have  (n d-1 ) geometric permutations in the worst case  same bound for connected components, previous was O(n 2d-4 ); also better bound in R 3  Hadwiger-type theorem with constant ≤ 2d  But no Helly-type! (need constant bound on geometric permutations)  Hadwiger-type theorem with constant ≤ 2d  But no Helly-type! (need constant bound on geometric permutations)

Conclusions and perspectives  Disjoint balls are nice wrt line transversals!  … but undoubtedly exceptions  Disjoint balls are nice wrt line transversals!  … but undoubtedly exceptions  Optimality (gap between lower and upper bounds)  congruent balls in 3D: Hadwiger between 5 and 6, Helly between 5 and 11  Number of geometric permutations of disjoint unit balls in R 3 : 2 or 3?  Algorithmic perspectives: GLP  Optimality (gap between lower and upper bounds)  congruent balls in 3D: Hadwiger between 5 and 6, Helly between 5 and 11  Number of geometric permutations of disjoint unit balls in R 3 : 2 or 3?  Algorithmic perspectives: GLP Thanks for your attention!