Points with large quadrant depth

Given a set <em>P</em> 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 <em>P</em>. We deal with an extremal version of this problem. A pair (<em>a</em>,<em>b</em>) of numbers is admissible if every point set <em>P</em> contains a point <em>p</em> in <em>P</em> that determines a pair (<em>Q</em>,<em>Q</em>^op) of opposite quadrants, such that<em>Q</em> contains at least an <em>a</em>-fraction and <em>Q</em>^op contains at least a <em>b</em>-fraction of the points of <em>P</em>. We provide a complete description of the set <em>F</em> of all admissible pairs (<em>a</em>,<em>b</em>). This amounts to identifying three line segments and a point on the boundary of <em>F</em>.<p> </p><p>In higher dimensions we study the maximum <em>a</em>, such that (<em>a</em>,<em>a</em>) is opposite-orthant admissible. In dimension <em>d</em> we show that 1/(2γ)≤<em>a</em>≤1/γ for γ=2^{2^<em>d</em>-1}2^<em>d</em>-1.</p><p>Finally we deal with a variant of the problem where the opposite pairs of orthants need not be determined by a point in <em>P</em>. Again we are interested in values <em>a</em>, such that all subsets <em>P</em> in<strong>R</strong>^d admit a pair (<em>O</em>,<em>O</em>^op) of opposite orthants both ontaining at least an <em>a</em>-fraction of the points. The maximum such value is <em>a</em>=1/2^<em>d</em>. Generalizations of the problem are also disussed.