Quadrilaterals inscribed in convex curves

We classify the set of quadrilaterals that can be inscribed in convex Jordan curves, continuous as well smooth case. This answers a question Makeev special case curves. The difficulty this problem comes from fact standard topological arguments to prove existence solutions do not apply here due lack sufficient symmetry. Instead, proof makes use an area argument Karasev and Tao, which we furthermore simplify elaborate on. requires additional analysis singular points, small miracle, then extends show problems inscribing isosceles trapezoids curves piecewise $C^1$ are equivalent.