Convex Quadrilaterals and k-Sets

  title={Convex Quadrilaterals and k-Sets},
  author={L{\'a}szl{\'o} Lov{\'a}sz and Katalin Vesztergombi and Uli Wagner and Emo Welzl},
We prove that the minimum number of convex quadrilaterals determined by n points in general position in the plane – or in other words, the rectilinear crossing number of the complete graph Kn – is at least (8 + 10 −5) (n 4 ) + O(n3). This is closely related to the rectilinear crossing number of complete graphs and to Sylvester’s Four Point Problem from the theory of geometric probabilities. As our main tool, we prove a lower bound on the number of (≤ k)-sets of the point set: for every k ≤ n/2… CONTINUE READING

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