Let be a simplicial triangulation of the 2-sphere, X the associated integral 2-cycle. A filling of X is an integral 3-chain Y with Y = X; a taut filling is one with minimal L₁-norm. We show that any taut filling arises from an extension of to a shellable simplicial triangulation of the 3-ball. The key to the proof is the general fact that any taut filling of an n-cycle splits under disjoint union, connected sum, and more generally what we call almost disjoint union, where summands are supported on sets that overlap in at most n+1 vertices. Despite the generality of this result, we have nothing to say about optimal fillings of spheres of dimension 3 or higher.
Doyle et al. (Wed,) studied this question.