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On a conjecture by Eckhoff and Dolnikov concerning line transversals to Euclidean disks

Wednesday, December 13th, 2017, 16:10

Schreiber 309

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On a conjecture by Eckhoff and Dolnikov concerning line transversals to Euclidean disks

Alexander Magazinov, TAU

Abstract:

The following problem was proposed in 1969 by Eckhoff and, independently, in 1972 by Dolnikov: -- Let a family $F$ of unit disks in the plane be given such that every three disks can be stabbed by one line. What is the minimum $\lambda$ for which we can guarantee that blowing up each disk from $F$ by a coefficient $\lambda$ yields a family of disks that can all be stabbed by a single line? -- Surprisingly, the problem is still open (Conjecture 4.2.25 in the upcoming edition of Handbook of Discrete and Computational Geometry). The conjectured value of $\lambda$ is the golden ratio $(\sqrt{5} + 1) / 2 \approx 1.618$. This is also the lower bound, since one can place the centers of five unit disks in the vertices of a regular pentagon of an appropriate size. In this talk the speaker will propose a (conjectural) statement which implies the Dolnikov--Eckhoff conjecture and is algebraic in an alphabet of 7 variables. The speaker managed to verify that statement on a sufficiently dense grid and thus obtain an estimate $\lambda < 1.645$, which is a significant improvement on the previous upper bound $\lambda < (1 + \sqrt{1 + 4 \sqrt{2}}) / 2 \ approx 1.790$ by Jeronimo Castro and Roldan-Pensado (2011).

 

 

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