Abstract We prove a fractional Helly theorem for k -flats intersecting fat convex sets. A family F F of sets is said to be 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. We prove that for every dimension d and positive reals ρ and α there exists a positive = (d, , ) β = β (d, ρ, α) such that if F F is a finite family of ρ -fat convex sets in Rᵈ R d and an 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 -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.
Jung et al. (Tue,) studied this question.