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Publications (5)
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 oppos...
Let P be a set of n points in general position in the plane. For every x∈P let D(x,P) be the maximum number such that there are at least D(x,P) points of P in each of two opposite quadrants determined by some two perpendicular lines through x. Define D(P)=maxx∈PD(x,P). In this paper we show that D(P)⩾c|P| for every set P in general position in the...
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 the extremal version of this problem. A pair (a, b) of numbers is admissible if every point set P contains a point p ∈ P that determines a pair (Q,Qop) of oppos...
We show that for every ϵ>0 there exists an angle α=α(ϵ) between 0 and π, depending only on ϵ, with the following two properties: (1) For any continuous probability measure in the plane one can find two lines ℓ1 and ℓ2, crossing at an angle of (at least) α, such that the measure of each of the two opposite quadrants of angle π−α, determined by ℓ1 an...
Liver tumors segmentation is an important prerequisite for planning of surgical interventions. For clinical applicability, the segmentation approach must be able to cope with the high variation in shape and gray-value appearance of the liver. We present a fully automatic 3D segmentation method for the liver tumors from contrast-enhanced CT data. Th...