TY - JOUR
T1 - Positive-fraction intersection results and variations of weak epsilon-nets
AU - Magazinov, Alexander
AU - Soberón, Pablo
N1 - Publisher Copyright:
© 2016, Springer-Verlag Wien.
PY - 2017/5/1
Y1 - 2017/5/1
N2 - Given a finite set X of points in Rn and a family F of sets generated by the pairs of points of X, we determine volumetric and structural conditions for the sets that allow us to guarantee the existence of a positive-fraction subfamily F′ of F for which the sets have non-empty intersection. This allows us to show the existence of weak epsilon-nets for these families. We also prove a topological variation of the existence of weak epsilon-nets for convex sets.
AB - Given a finite set X of points in Rn and a family F of sets generated by the pairs of points of X, we determine volumetric and structural conditions for the sets that allow us to guarantee the existence of a positive-fraction subfamily F′ of F for which the sets have non-empty intersection. This allows us to show the existence of weak epsilon-nets for these families. We also prove a topological variation of the existence of weak epsilon-nets for convex sets.
KW - Positive fraction intersection
KW - Selection theorem
KW - Weak epsilon-nets
UR - http://www.scopus.com/inward/record.url?scp=84961150743&partnerID=8YFLogxK
U2 - 10.1007/s00605-016-0892-2
DO - 10.1007/s00605-016-0892-2
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AN - SCOPUS:84961150743
SN - 0026-9255
VL - 183
SP - 165
EP - 176
JO - Monatshefte fur Mathematik
JF - Monatshefte fur Mathematik
IS - 1
ER -