J. Matousek. Geometric set systems. In European Congress of Mathematics, Vol. II (Budapest 1996), volume 169 of Progr. Math., pages 1--27. Birkhauser, Basel, 1998.  Home/Search Document Details and Download
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Clustering for Edge-Cost Minimization - Schulman (14 citations) (Correct)
.... and Shelah [64] showed that the number of subsets of a set of size n that can be defined by intersections with elements of a range space of VC dimension d, is at most Phi d (n) For a discussion of applications in learning theory, statistics, and combinatorial and computational geometry see [43, 70, 52]. Dudley has shown that the range space of balls in dimension d has VC dimension d 1 [17] and the bound Phi d 1 (n) follows. 2 The 2 stable partitions are even further restricted: Proposition 10 If S ae R then j(R) ae j(S) Furthermore (R) S) can contain at most one point. Proof: A ....
....2 partitions. To begin with, a o(n d 1 ) bound on the size of an antichain in the poset of intersections with spheres would be required; the existence of such a bound seems to be an open question. The related k sets question for intersections with halfspaces is a long standing challenge; see [52, 49, 22, 19, 5, 3, 57, 21, 15]. It would also be necessary to implement an efficient search of sphere partitions which did not expend much effort on spheres not in the relevant 2 family. 3.5 Exact deterministic algorithm for k partitions Consider a 1 stable k partition. Just as for 2 partitions, space is partitioned into k ....
J. Matousek. Geometric set systems. to appear in Proc. 2'nd European Math. Congress.
Mathematical Snapshots From the Computational Geometry Landscape - Matousek (Correct)
....line crossing O( p r) parts only (see [25] Both these results are asymptotically optimal. The research in range searching also initiated a fruitful theory related to the so called Vapnik Chervonenkis dimension of set systems, with applications, e.g. in discrepancy theory; this is surveyed in [27]. Lower bounds for range searching were proved mainly by Chazelle; a key paper is [9] In the proof, some integral geometric considerations appear, and, interestingly, the lower bounds are related to a generalization of Heilbronn s problem from discrete geometry. For an n point set P ae [0; 1] ....
J. Matousek. Geometric set systems. In Proceedings of the 2nd European Congress of Mathematicians. Birkhauser, Basel, 1998. In press.
Discrete Mathematics and Theoretical Computer Science DMTCS.. - Bernard Ycart And (Correct)
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J. Matousek. Geometric set systems. In European Congress of Mathematics, Vol. II (Budapest 1996), volume 169 of Progr. Math., pages 1--27. Birkhauser, Basel, 1998.
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