Fractional Helly Theorem for Cartesian Products of Convex Sets

Abstract

Helly's theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by Barany and Kalai, and independently Lew, we generalize Eckhoff's result to Cartesian products of convex sets in all dimensions. In particular, we prove that given alpha is an element of(1-1/t(d),1] and a finite family F of Cartesian products of convex sets Pi(i is an element of[t]) A(i) in R-td with A(i) subset of R-d, if at least alpha-fraction of the (d+1)-tuples in F are intersecting, then at least (1-(td(1-alpha))(1/(d+1)))-fraction of sets in F are intersecting. This is a special case of a more general result on intersections of d-Leray complexes. We also provide a construction showing that our result on d-Leray complexes is optimal. Interestingly, the extremal example is representable as a family of Cartesian products of convex sets, implying that the bound alpha>1-1/t(d) and the fraction (1-(t(d)(1-alpha))(1/(d+1))) above are also best possible. The well-known optimal construction for fractional Helly theorem for convex sets in R-d does not have (p,d+1)-condition for sublinear p, that is, it contains a linear-size subfamily with no intersecting (d+1)-tuple. Inspired by this, we give constructions showing that, somewhat surprisingly, imposing an additional (p,d+1)-condition has a negligible effect on improving the quantitative bounds in neither the fractional Helly theorem for convex sets nor Cartesian products of convex sets. Our constructions offer a rich family of distinct extremal configurations for fractional Helly theorem, implying in a sense that the optimal bound is stable.