We discuss two variational problems (denote them P1, P2) that encode the idea of ``optimally'' approximating an n-d measure \rho by distorting an m-d set (m < n). Intuitively, when n=3 and m=1, the loose idea is to do something like tying a balloon dog, while for n=3 and m=2, we may think of something more like origami.

We formalize these ideas in terms of minimizing an average distance from \rho to an approximating m-d set, subject to a complexity constraint. The way that we choose to quantify the complexity dictates the problem's behavior. In (P1) we use a W^{k,q} (kq > m) Sobolev norm of a parametrization; this is basically a principal manifold problem with higher-order regularization of data noise. In (P2) we restrict to m=1 and use the Hausdorff 1-measure; this yields the celebrated ``average distance problem,'' which is closely related to (but distinct from) the problem recently studied by Chambolle, Duval, and Machado. In both (P1) and (P2) the objective functional has a simple OT description; we discuss its ``gradient.'' Using it, in (P2) we are able to mostly resolve an open question about whether optimizers must be finite binary trees. The analogous question was listed as open in the Chambolle et. al framework; it is possible our methods could be adapted to this context.

P1 is joint work with Jonathan Hayase and Young-Heon Kim; P2 is joint work with Lucas O'Brien and Young-Heon Kim.