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 Bárány 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 $$\alpha \in (1-1/t^d,1]$$ and a finite family $${\mathcal {F}}$$ of Cartesian products of convex sets $$\prod _{i\in [t]}A_i$$ in $${\mathbb {R}}^{td}$$ with $$A_i\subset {\mathbb {R}}^d$$ , if at least $$\alpha $$ -fraction of the $$(d+1)$$ -tuples in $${\mathcal {F}}$$ are intersecting, then at least $$(1-(t^d(1-\alpha ))^{1/(d+1)})$$ -fraction of sets in $${\mathcal {F}}$$ 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 $${\mathbb {R}}^d$$ 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.