  We study the interplay between the recently defined concept of minimum
homotopy area and the classical topic of self-overlapping curves. The latter
are plane curves which are the image of the boundary of an immersed disk. Our
first contribution is to prove new sufficient combinatorial conditions for a
curve to be self-overlapping. We show that a curve $\gamma$ with Whitney index
1 and without any self-overlapping subcurves is self-overlapping. As a
corollary, we obtain sufficient conditions for self-overlappingness solely in
terms of the Whitney index of the curve and its subcurves. These results follow
from our second contribution, which shows that any plane curve $\gamma$, modulo
a basepoint condition, is transformed into an interior boundary by wrapping
around $\gamma$ with Jordan curves. Equivalently, the minimum homotopy area of
$\gamma$ is reduced to the minimal possible threshold, namely the winding area,
through wrapping. In fact, we show that $n+1$ wraps suffice, where $\gamma$ has
$n$ vertices. Our third contribution is to prove the equivalence of various
definitions of self-overlapping curves and interior boundaries, often implicit
in the literature. We also introduce and characterize zero-obstinance curves,
further generalizations of interior boundaries defined by optimality in minimum
homotopy area.
