TY - JOUR
T1 - Bounding the piercing number
AU - Alon, N.
AU - Kalai, G.
PY - 1995/12
Y1 - 1995/12
N2 - It is shown that for every k and every p≥q≥d+1 there is a c=c(k,p,q,d)<∞ such that the following holds. For every family ℋ whose members are unions of at most k compact convex sets in R d in which any set of p members of the family contains a subset of cardinality q with a nonempty intersection there is a set of at most c points in R d that intersects each member of ℋ. It is also shown that for every p≥q≥d+1 there is a C=C(p,q,d)<∞ such that, for every family[Figure not available: see fulltext.] of compact, convex sets in R d so that among and p of them some q have a common hyperplane transversal, there is a set of at most C hyperplanes that together meet all the members of[Figure not available: see fulltext.].
AB - It is shown that for every k and every p≥q≥d+1 there is a c=c(k,p,q,d)<∞ such that the following holds. For every family ℋ whose members are unions of at most k compact convex sets in R d in which any set of p members of the family contains a subset of cardinality q with a nonempty intersection there is a set of at most c points in R d that intersects each member of ℋ. It is also shown that for every p≥q≥d+1 there is a C=C(p,q,d)<∞ such that, for every family[Figure not available: see fulltext.] of compact, convex sets in R d so that among and p of them some q have a common hyperplane transversal, there is a set of at most C hyperplanes that together meet all the members of[Figure not available: see fulltext.].
UR - http://www.scopus.com/inward/record.url?scp=51249163109&partnerID=8YFLogxK
U2 - 10.1007/BF02574042
DO - 10.1007/BF02574042
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AN - SCOPUS:51249163109
SN - 0179-5376
VL - 13
SP - 245
EP - 256
JO - Discrete and Computational Geometry
JF - Discrete and Computational Geometry
IS - 1
ER -