arXiv subjects

Tobias Christ

Publications and source records attributed to Tobias Christ.

2 recordsLinked to original sources

arXiv2011

A Doubly Exponentially Crumbled Cake

We consider the following cake cutting game: Alice chooses a set P of n points in the square (cake) [0,1]^2, where (0,0) is in P; Bob cuts out n axis-parallel rectangles with disjoint interiors, each of them having a point of P as the lower left corner; Alice keeps the rest. It has been conjectured that Bob can always secure at least half of the cake. This remains unsettled, and it is not even known whether Bob can get any positive fraction independent of n. We prove that if Alice can force Bob's share to tend to zero, then she must use very many points; namely, to prevent Bob from gaining more than 1/r of the cake, she needs at least 2^{2^{Ω(r)}} points.

cs.CG↗

arXiv2010

Consistent digital line segments

We introduce a novel and general approach for digitalization of line segments in the plane that satisfies a set of axioms naturally arising from Euclidean axioms. In particular, we show how to derive such a system of digital segments from any total order on the integers. As a consequence, using a well-chosen total order, we manage to define a system of digital segments such that all digital segments are, in Hausdorff metric, optimally close to their corresponding Euclidean segments, thus giving an explicit construction that resolves the main question of Chun et al.

cs.DM↗