  In 1974, Gehring posed the problem of minimizing the length of two linked
curves separated by unit distance. This constraint can be viewed as a measure
of thickness for links, and the ratio of length over thickness as the
ropelength. In this paper we refine Gehring's problem to deal with links in a
fixed link-homotopy class: we prove ropelength minimizers exist and introduce a
theory of ropelength criticality.
  Our balance criterion is a set of necessary and sufficient conditions for
criticality, based on a strengthened, infinite-dimensional version of the
Kuhn--Tucker theorem. We use this to prove that every critical link is C^1 with
finite total curvature. The balance criterion also allows us to explicitly
describe critical configurations (and presumed minimizers) for many links
including the Borromean rings. We also exhibit a surprising critical
configuration for two clasped ropes: near their tips the curvature is unbounded
and a small gap appears between the two components. These examples reveal the
depth and richness hidden in Gehring's problem and our natural extension.
