According to the Boros-F\"uredi-B\'ar\'any theorem, for any system of n points in Euclidean d-space, there is a point contained in at least a constant fraction of all simplices generated by them. We discuss some related problems for geometric arrangements. One of our tools will be the following: Let α>0, let H1,…,Hm be finite families of semi-algebraic sets of constant

description complexity, and let R be a fixed semi-algebraic m-ary relation on H1×⋯×Hm such that the number of m-tuples that are related (resp. unrelated) with respect to R is at least α∏mi=1∣∣Hi∣∣. Then there exists a constant c>0, which depends on α,m and on the maximum description complexity of the sets in Hi(1≤i≤m) and R, and exist subfamilies H∗i⊆Hi with ∣∣H∗i∣∣≥c′∣∣Hi∣∣(1≤i≤m) such that H∗1×⋯×H∗m⊂R (resp. H∗1×⋯×H∗m∩R=∅). (Joint work with Jacob Fox, Mikhail Gromov, Vincent Lafforgue \and Assaf Naor.)