line transversals to disjoint balls (joint with c. borcea, o. cheong, x. goaoc, a. holmsen) sylvain...

Line Transversals to Disjoint BallsLine Transversals to Disjoint Balls

(Joint with C. Borcea, O. Cheong, X. Goaoc, A. Holmsen)

(Joint with C. Borcea, O. Cheong, X. Goaoc, A. Holmsen)

Sylvain Petitjean

LORIA, Vegas team

O. Schwarzkopf

A long time ago…A long time ago…

E. Helly (1923): n convex sets in Rd 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 Rd have a point in common iff every d+1 have a point in common.Basic combinatorial result on convex sets

TransversalsTransversals

Reformulate: points hitting convex sets 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.

MilestonesMilestones Danzer (1957): n disjoint unit discs in R2 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 R2 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 R2 have a line transversal if and only if every triple has a transversal consistent with some fixed orderorder is important

Hadwiger (1957): n disjoint convex sets in R2 have a line transversal if and only if every triple has a transversal consistent with some fixed orderorder is important

2

31

4

€

2 p 3 p 1 p 4

In 3D: bummer again!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)

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?What about balls?

Danzer’s conjecture: Helly for disjoint balls in nD

Danzer’s conjecture: Helly for disjoint balls in nD

type Hadwiger number Helly number

Hadwiger (1957) & Grünbaum (1960)

thinly distributed in Rd d2 2d-1 d2 2d-1

Holmsen et al. (2003) & Cheong et al. (2005)

disjoint unit in R3 12 6 46 11

Cheong et al. (2006) pairwise-inflatable in Rd 2d 4d-1

Borcea et al. (2007) disjoint in Rd 2d

€

dij > 2(ri + rj )

€

dij2 > 2(ri

2 + rj2)

Convexity of cone of directionsConvexity of cone of directions

Borcea, Goaoc, P. (2007): Directions of oriented lines stabbing a finite family of disjoint balls in Rd in a given order form a strictly convex subset of Sd-1

Borcea, Goaoc, P. (2007): Directions of oriented lines stabbing a finite family of disjoint balls in Rd in a given order form a strictly convex subset of Sd-1

Instrumental in most proofs in transversal theory Instrumental in most proofs in transversal theory

Previously known for thinly distributed balls (Hadwiger), pairwise-inflatable balls

Previously known for thinly distributed balls (Hadwiger), pairwise-inflatable balls

3D case: 3 disjoint balls3D case: 3 disjoint balls

New proof techniqueNew proof technique

Write down equations conics and sextic

Write down equations conics and sextic

Identify the border arcs Identify the border arcs

Prove Hessian does not meet them local convexity

Prove Hessian does not meet them local convexity

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Argue that cone is contractible Argue that cone is contractibleQuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.

Disjointness is a natural boundaryDisjointness is a natural boundary

QuickTime™ and aTIFF (LZW) decompressor






Extension to higher dimensionsExtension to higher dimensions



Implications: disjoint ballsImplications: disjoint balls

Isotopy = geometric permutations Smorodinsky et al. (2000): n disjoint balls in Rd

have (nd-1) geometric permutations in the worst case same bound for connected components, previous was

O(n2d-4); also better bound in R3

Isotopy = geometric permutations Smorodinsky et al. (2000): n disjoint balls in Rd

have (nd-1) geometric permutations in the worst case same bound for connected components, previous was

O(n2d-4); also better bound in R3

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 perspectivesConclusions 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 R3: 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 R3: 2 or 3?

Algorithmic perspectives: GLP

Thanks for your attention!

line transversals to disjoint balls (joint with c. borcea, o. cheong, x. goaoc, a. holmsen) sylvain...

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