Hadwiger and Helly-type theorems for disjoint unit spheres in R3

Let S be an ordered set of disjoint unit spheres in R3 We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for every subset of at most 11 spheres from S implies the existence of a line transversal forS.

[1]  Tom C. Brown.  Common Transversals , 1976, J. Comb. Theory, Ser. A.

[2]  János Pach,et al.  Common Tangents to Four Unit Balls in R3 , 2001, Discret. Comput. Geom..

[3]  L. Danzer.  Über ein Problem aus der kombinatorischen Geometrie , 1957 .

[4]  E. Helly,et al.  Über Systeme von abgeschlossenen Mengen mit gemeinschaftlichen Punkten , 1930 .

[5]  JE MET,et al.  Wiskundige Opgaven , 1892 .

[6]  Helge Tverberg,et al.  Proof of grünbaum's conjecture on common transversals for translates , 1989, Discret. Comput. Geom..

[7]  H. Debrunner,et al.  Helly Type Theorems Derived from Basic Singular Homology , 1970 .

[8]  Nina Amenta,et al.  Helly-type theorems and Generalized Linear Programming , 1994, Discret. Comput. Geom..

[9]  J. Matousek,et al.  Using The Borsuk-Ulam Theorem , 2007 .

[10]  Subhash Suri,et al.  Geometric permutations of balls with bounded size disparity , 2001, Comput. Geom..

[11]  M. Katchalski.  A conjecture of Grünbaum on common transversals. , 1986 .

[12]  Ferenc Fodor,et al.  A Helly-type transversal theorem for n-dimensional unit balls , 2006 .

[13]  Rephael Wenger,et al.  Helly-Type Theorems and Geometric Transversals , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[14]  E. Helly.  Über Mengen konvexer Körper mit gemeinschaftlichen Punkte. , 1923 .

[15]  B. Grünbaum.  Common Transversals for Families of Sets , 1960 .

[16]  Xavier Goaoc,et al.  Geometric permutations of disjoint unit spheres , 2005, Comput. Geom..

[17]  R. Pollack,et al.  Geometric Transversal Theory , 1993 .

[18]  Subhash Suri,et al.  A Constant Bound for Geometric Permutations of Disjoint Unit Balls , 2003, Discret. Comput. Geom..

[19]  Ted G. Lewis,et al.  A Helly-Type Theorem for Line Transversals to Disjoint Unit Balls , 2003, Discret. Comput. Geom..

[20]  Jirí Matousek,et al.  No Helly Theorem for Stabbing Translates by Lines in R3 , 2004, Discret. Comput. Geom..

[21]  H. Hadwiger.  Ueber Eibereiche mit gemeinsamer Treffgeraden , 1957 .

[22]  Frank Sottile,et al.  The Envelope of Lines Meeting a Fixed Line and Tangent to Two Spheres , 2005, Discret. Comput. Geom..