Would a proof of Convex Quadrilaterals Inscribed In Jordan Curves be interesting? I know that there is a result that shows that every Jordan curve always has an inscribed rectangle without defined aspect ratio.
And of course the open Square Peg Problem asks whether there is an inscribed square for every Jordan curve.
With the realization that every rectangle is a convex quadrilateral, would it still be interesting to prove that every Jordan curve has an inscribed convex quadrilateral?  
I have a proof for this.
 A: May I recommend that you get in touch with Elizabeth Denne, who is in some sense the world-expert in this area?
There is so much interest in 
the inscribed square question
that it is difficult to contribute anything new unless you have mastered all
that has preceded.
A: Here is the very brief proof for anyone interested:
Lemma 1:(Meyerson,1980) If J is any simple closed curve and T is any triangle then J has an inscribed triangle similar to T.
Claim: Every Jordan curve has an inscribed convex quadrilateral.
Proof of Claim: 
Let γ be a continuous closed curve in the plane without self-intersections and assume that γ has no inscribed convex quadrilaterals. 
By Lemma 1, J has an inscribed equilateral triangle ABC.
Thus, there is a Jordan arc J1 joining A and B and a Jordan arc J2 joining A and C. 
A line L can be drawn through the center of ABC parallel to the segment BC such that some two points, m1 on J1 and m2 on J1 lie on L.
Thus, points m1,m2,B, and C form the inscribed convex quadrilateral m1m2BC.
Q.E.D
