Bounded flat transversals for fat convex bodies

Fix \(d\geq3\), \(1\leq k\leq d-2\), \(p\geq k+2\), and a fatness parameter \(\rho\geq1\). We prove that a finite family of \(\rho\)-fat compact convex bodies in \(\R^d\) can be intersected by a bounded number of affine \(k\)-flats whenever every \(p\) members contain \(k+2\) members with a common \(k\)-flat transversal. The bound depends only on \(d,k,p,\rho\); the bodies may have arbitrary sizes and convex boundaries. The missing rounding step is obtained by grouping bodies around members of minimum radius, then reducing the dimension of their transversals on the sphere. A finite family of perturbations turns the resulting approximate transversals into exact ones at the scale of the smallest body in each group.