  Consider an inextensible closed filament immersed in a 2D Stokes fluid. Given
a force density $\mathbf{F}$ defined on this filament, we consider the problem
of determining the tension $\sigma$ on this filament that ensures the filament
is inextensible. This is a subproblem of dynamic inextensible vesicle and
membrane problems, which appear in engineering and biological applications. We
study the well-posedness and regularity properties of this problem in H\"older
spaces. We find that the tension determination problem admits a unique solution
if and only if the closed filament is {\em not} a circle. Furthermore, we show
that the tension $\sigma$ gains one derivative with respect to the imposed line
force density $\mathbf{F}$, and show that the tangential and normal components
of $\mathbf{F}$ affect the regularity of $\sigma$ in different ways. We also
study the near singularity of the tension determination problem as the
interface approaches a circle, and verify our analytical results against
numerical experiment.
