Compact families of Jordan curves and convex hulls in three dimensions

Abstract

We prove that for certain families of semi-algebraic convex bodies in 3 dimensions, the convex hull of $n$ disjoint bodies has $O(n\lambda_s(n))$ features, where $s$ is a constant depending on the family: $\lambda_s(n)$ 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' of disc intersections.