k-Dimensional Transversals For Fat Convex Sets

We prove a fractional Helly theorem for k -flats intersecting fat convex sets. A family F F of sets is said to be ρ ρ -fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii of these balls is bounded by ρ ρ . We prove that for every dimension d and positive reals ρ ρ and α α there exists a positive β =β (d,ρ , α ) β = β ( d , ρ , α ) such that if F F is a finite family of ρ ρ -fat convex sets in Rᵈ R d and an α α -fraction of the $$(k+2)$$ ( k + 2 ) -size subfamilies from F F can be hit by a k -flat, then there is a k -flat that intersects at least a β β -fraction of the sets of F F . We prove spherical and colorful variants of the above results and prove a $$(p,k+2)$$ ( p , k + 2 ) -theorem for k -flats intersecting balls.