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Convex polygons in density-restricted point sets
Zálešák, Ondřej ; Valtr, Pavel (advisor) ; Balko, Martin (referee)
For A, a finite set of points in Rd , let ∆(A) denote the spread of A and be equal to the ratio of the maximum and the minimum distance of two points from A. Valtr (1992) proved that for sets of points in the plane with spread equal to Θ(n 1 2 ), the cardinality of the largest subset in convex position is Θ(n 1 3 ) in the worst case. The same article also contains an expanded upper bound on the guaranteed cardinality of subsets in convex position for sets with spread asymptotically higher than n 1 2 , and a brief construction for the proof. This thesis looks at this construction in detail. Furthermore it builds on the recent results for sets in higher dimensions, specifically discusses, whether it is possible to expand the upper bound in three-dimensional space for higher spreads with a similar technique as in the planar case. 1 Seznam použité literatury Valtr, P. (1992). Convex independent sets and 7-holes in restricted planar point sets. Discrete & Computational Geometry, 7(2), 135-152. 2
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