### Points with large quadrant depth

#### Abstract

Given a set

*P*of points in the plane we are interested in points that are `deep' in the set in the sense that they have two opposite quadrants both containing many points of*P*. We deal with an extremal version of this problem. A pair (*a*,*b*) of numbers is admissible if every point set*P*contains a point*p*in*P*that determines a pair (*Q*,*Q*^{op}) of opposite quadrants, such that*Q*contains at least an*a*-fraction and*Q*^{op}contains at least a*b*-fraction of the points of*P*. We provide a complete description of the set*F*of all admissible pairs (*a*,*b*). This amounts to identifying three line segments and a point on the boundary of*F*.

In higher dimensions we study the maximum *a*, such that (*a*,*a*) is opposite-orthant admissible. In dimension *d* we show that 1/(2γ)≤*a*≤1/γ for γ=2^{2d-1}2^{d-1}.

Finally we deal with a variant of the problem where the opposite pairs of orthants need not be determined by a point in *P*. Again we are interested in values *a*, such that all subsets *P* in**R**^{d} admit a pair (*O*,*O*^{op}) of opposite orthants both ontaining at least an *a*-fraction of the points. The maximum such value is *a*=1/2^{d}. Generalizations of the problem are also disussed.

#### Full Text:

PDFDOI: http://dx.doi.org/10.20382/jocg.v2i1a7

ISSN: 1920-180X