Is this a counter example to Stromquist's Theorem? Stromquist's Theorem:  If the simple closed curve J is "nice enough" then it has an inscribed square.
"Nice enough" includes polygons.
Read more about it here:  www.webpages.uidaho.edu/~markn/squares
An "inscribed square" means that the corners of a square overlap with the curve.
I would like to suggest a counter-example:
The curve connected by the points$$
(0.2,0),\    (1,0),\ (1,1),\ (0,1),\ (0,0.2),\ (-0.2, -0.2),\ (0.2,0).$$
Link to plot: http://www.freeimagehosting.net/uploads/5b289e6824.png
Can this curve be incribed by a square?
(An older version of this question had another example: a triangle on top of a square (without their mutual side.) )

Regarding your edit: (0.2, 0) — (1, 0.2) — (0.8, 1) — (0, 0.8) (and many others)
http://www.imgftw.net/img/326639277.png
