$k$-dimensional transversals for fat convex sets

We prove a fractional Helly theorem for $k$-flats intersecting fat convex sets. A family $\mathcal{F}$ of sets is said to be $\rho$-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 $\rho$. We prove that for every dimension $d$ and positive reals $\rho$ and $\alpha$ there exists a positive $\beta=\beta(d,\rho, \alpha)$ such that if $\mathcal{F}$ is a finite family of $\rho$-fat convex sets in $\mathbb{R}^d$ and an $\alpha$-fraction of the $(k+2)$-size subfamilies from $\mathcal{F}$ can be hit by a $k$-flat, then there is a $k$-flat that intersects at least a $\beta$-fraction of the sets of $\mathcal{F}$. We prove spherical and colorful variants of the above results and prove a $(p,k+2)$-theorem for $k$-flats intersecting balls.