The claim I'd like to prove or disprove is in the title. Here, convexity means every tangent line at any point on the curve has the whole curve in one half of it. The curve being closed means that it's an image of some parametrization, such that the starting and end point of the parametrization domain get mapped to the same point in $\mathbb{R^2}$.
An idea someone gave me for proving the claim - look at tangential/non-tangential intersections (at presumed points of intersection in the curve) - in the non-tangential scenario I think I get the idea, but I wasn't quite sure how it would be shown in the case of an tangential intersection.
An idea for proving the claim:
If $C$ is not a Jordan curve, then there has to be a Jordan curve (as a subset in the curve), such that this Jordan curve /loop (I'll refer to it as a loop from now) is not the whole curve $C$ - not sure how I would prove this, I'll call this statement $*$. If the whole curve $C$ is convex, then any subset, and thus this loop as well, is convex.
So there's some point of the curve that isn't in this loop we've found:
If it's inside the loop, then the tangent halves the loop, which is a contradiction.
If the point is outside, take the closest point on the loop (exists from compactness) - now the tangent at this point has the outside point on one half, and the loop on the other (not sure how you would show this, but it should be true for something with an interior set like loop), which is a contradiction.
Therefore, the curve is a Jordan curve.
EDIT:
Idea on how to prove $*$:
Using an arclenght parametrization $c$, the length of the domain of $c$ is finite, the curvature is bounded, and so if I consider any point of intersection, and some two points in the domain that get mapped to this intersection point, the total curvature between them is at least 2$\pi$, and so the distance between the points has a lower bound.
Therefore there's finitely many points in the domain that map to a point of intersection, and so we can find two points $a$ and $b$ that map to a point of intersection, such that there's no two or more points between $a$ and $b$ that would map to some point of intersection - we've found a loop/ Jordan curve 'inside' $C$.