Sharp fractional Helly bounds for common affine flats

If an $\alpha$ fraction of the $(d-k+1)$-tuples of closed convex sets in $\mathbb R^d$ have an intersection containing a whole affine $k$-flat, then a fraction $1-(1-\alpha)^{1/(d-k+1)}$ of the sets share such a flat. This coefficient is best possible, resolving the sharp fractional De Santis question of Ludwigson and Soberón. The geometric result behind the bound is $(d-k)$-collapsibility of the common-flat complex, proved by coordinating convex sweeps in varying quotient spaces.