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- W2293635260 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 2 d -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." @default.
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- W2293635260 date "2011-10-01" @default.
- W2293635260 modified "2023-10-14" @default.
- W2293635260 title "Points with large quadrant depth" @default.
- W2293635260 doi "https://doi.org/10.20382/jocg.v2i1a7" @default.
- W2293635260 hasPublicationYear "2011" @default.
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