Two equivalent definitions of a *convex curve* using curvature and set theory I have seen two different definitions of a closed convex curve in a plane:
For a curve $r\in\mathcal C^2([0,L],\mathbb R^2)$,


*

*The whole curve lies on only one side of any tangent of the curve 

*Any straight line (geodesic) joining two points within the region bounded by the curve lies within the region. I.e. the curve is the topological boundary of a convex set.
How can I prove that both definitions are equivalent?
 A: You can use the following explanation. Here, $\theta$ denotes the angle from the $x$-axis to the tangent of the curve.
A positive $2\pi$-periodic function represents the curvature function of a simply closed, strictly convex $C^{2}$ plane curve if and only if 
$$\int_{0}^{2\pi}\displaystyle\frac{\cos \theta}{k(\theta)} \,\mathrm d\theta=\int_{0}^{2\pi}\displaystyle\frac{\sin \theta}{k(\theta)} \,\mathrm d\theta =0.$$
because if $k$ is the curvature function of some curve, then this relation follows directly from the fact that the curve is closed, i.e. that $\int_{0}^{L} T\,\mathrm ds=0$. 
In the other direction: Given an arbitrary $k$, the associated curve, up to translation, is defined by 
$$x(\theta)=\int_{0}^{\theta}\displaystyle\frac{\cos\eta}{k(\eta)}\,\mathrm d\eta\qquad y(\theta)=\int_{0}^{\theta}\displaystyle\frac{\sin\eta}{k(\eta)}\,\mathrm d\eta$$
It is easy to check that this curve is closed and simple as $x(0) = y(0) =
x(2\pi) = y(2\pi) = 0$.
A: This is just an idea. I focus first on your first statement. Suppose you parametrize r(t) by arc length with a positive 
orientation. Since a regular closed plane curve is convex if and only if it is simple and its curvature doesn't change sign, 
one can see that at any point p  of the curve r(t) where the curvature is negative in the neighbourhood of p the tangent line
to r(t) at p is in the interior of r(t). Since r(t) is bounded and the tangent is not, the tangent must intersect your curve 
at another point. A similar argument applies when considering the second statement, i.e. a straight line (geodesic) joining two 
points of r(t) lies within the region delimited by the curve if and only if the curve is simple and its curvature doesn't 
change sign.
