A property of compact, connected, laminated subsets of manifolds

Let X be a compact, connected subset of a smooth manifold. Suppose that X admits a codimension d lamination, d greater than or equal to 2. Let x, y epsilon X and epsilon > 0. There exists a sequence x = x(0),...,x(k) = y in X such that the sum of the dth powers of the distances between successive points is less than c. We discuss two proposed applications of this result.