Convexity criterion for piecewise regular planar curves A theorem of classical differential geometry states, that a simple, closed and regular $\mathcal{C}^2$-curve $\gamma:[0;1]\to\mathbb{R}^2$ is convex, iff its signed curvature doesn't change sign.
Assume now you have a simple, closed curve $\gamma:[0;1]\to\mathbb{R}^2$ and points
$0\leq t_0<t_1<...<t_n\leq 1$, so that $\gamma$ is regular and $\mathcal{C}^2$ on each interval $[0, t_0)$, $(t_0, t_1)$, ..., $(t_{n-1}, t_n)$, $(t_n, 1]$ and left- and right-sided differentiable at the points $t_0, ..., t_n$.
Is it true, that $\gamma:[0;1]\to\mathbb{R}^2$ is convex, iff its signed curvature doesn't change sign (where it is defined) and the inner angles at the "vertices" $t_0, ..., t_n$ are all less than $180^\circ$ ?
Or is there a similar criterion for convexity of piecewise regular, simple and closed planar curves?
 A: the thing you expect is true. I would just emphasize one little point, you cannot allow any self intersection, any two arcs, any arc through some endpoint not its own, coincidence of two endpoints. I guess you did cover that by saying simple closed curve...
This probably does it. At each vertex, round off the corner with a short almost-circular arc, making a $C^2$  curve that obeys the theorem. The original piecewise smooth body will be the union of an increasing infinite family of convex curves (including interiors), and convex. 
The annoying part is to show that, at any vertex, we can round the corner with some sort of spline arbitrarily close to the corner so as to have a $C^2$ curve.
A more general fact is called The Theorem of Turning Tangents. Let me try to find some references. 
Well, I'm not sure where you would find a careful proof of just the statement you made. Turning Tangents is both more and less than what you said. As I look at the extra condition I put, I wonder if this even requires the Jordan Curve Theorem. Hmmmm. Maybe not. You need just one point in the interior, around which the curve will wind. 
https://books.google.com/books?id=KUYLhOVkaV4C&pg=PA222&lpg=PA222&dq=theorem+of+turning+tangents&source=bl&ots=cEHiHlsWTa&sig=cgDItIad1v0iKS1vMEjA6IHYAjI&hl=en&sa=X&ved=0CEwQ6AEwBmoVChMI7sb5xtGGyQIVUy6ICh2PQQXg#v=onepage&q=theorem%20of%20turning%20tangents&f=false
