Compact families of Jordan curves and convex hulls in three dimensions

We prove that for certain families of semi-algebraic convex bodies in R3, the convex hull of n disjoint bodies has O(nλs(n)) features, where s is a constant depending on the family: λs(n) is the maximum length of order-s Davenport-Schinzel sequences with n letters. The argument is based on an apparently new idea of ‘compact families’ of convex bodies or discs, and of ‘crossing content’ and ‘footprint width’ among disc intersections.