Take a loop in the unit disk D^2, with length l where length is defined as the supremum of the lengths of piecewise linear approximations. What is the smallest r such that every radius-r subdisk of D^2 intersects the loop?

The question can of course be generalized by choosing different spaces to "fill", such as polygons, higher-dimensional balls D^n, various other compact manifolds etc. Having a path with free ends rather than a loop is another possible generalization. However, just the initial form of the question has me stumped. I don't know where to begin.

Note that generalizing the dimension of the "loop" does not yield interesting results, as r can be made arbitrarily small for fixed "length" l.