A New Lower Bound Based on Gromov's Method of Selecting Heavily Covered Points

Daniel Král' (1) , Lukáš Mach (1) , Jean-Sébastien Sereni (2)

1 DIMAP - Centre for Discrete Mathematics and its Applications [Warwick]
2 ORPAILLEUR - Knowledge representation, reasonning

Boros and Füredi (for d=2) and Bárány (for arbitrary d) proved that there exists a positive real number c_d such that for every set P of n points in R^d in general position, there exists a point of R^d contained in at least c_d * \binom{n}{d+1} d-simplices with vertices at the points of P. Gromov improved the known lower bound on c_d by topological means. Using methods from extremal combinatorics, we improve one of the quantities appearing in Gromov's approach and thereby provide a new stronger lower bound on c_d for arbitrary d. In particular, we improve the lower bound on c₃ from 0.06332 to more than 0.07480; the best upper bound known on c₃ being 0.09375.

Domaines: Combinatoire [math.CO]

Daniel Král', Lukáš Mach, Jean-Sébastien Sereni. A New Lower Bound Based on Gromov's Method of Selecting Heavily Covered Points. Discrete and Computational Geometry, 2012, 48 (2), pp.487--498. ⟨10.1007/s00454-012-9419-3⟩. ⟨hal-00912562⟩