The classical isoperimetric inequality states that a curve in the plane of length L bounds a disk whose area is at most L^2/4\pi. This inequality was generaized to curves in R^3 in the early 1900's. Such a curve bounds an immersed disk whose area is at most L^2/4\pi. It also bounds an embedded surface satisfying the same area bound.
An unknotted curve bounds an embedded disk in R^3. We show, in contrast to the above, that given any positive constant A, there are unknotted smooth curves of length 1 that do not bound embedded disks of area less than A. If we control the size of a tubular neighborhood of a curve then we do get explicit isoperimetric bounds.
(joint work with J. Lagarias and W. Thurston)
