# convex curve equivalent definition

I have seen two different version of definition of a closed convex curve in a plane:

For a curve $r(t)=(x(t),y(t))$

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

2.Any straight line (geodesic) joining two points within the region bounded by the curve lies within the region.

Obviously the two definitions are equivalent but I cannot prove. Can anybody give me some ideas?

u can use this explanation where $\theta$ is the angle from the $x$ axis to the tangent A positive $2\pi$ periodic function represents the curvature function of a simple closed strictly convex $C^{2}$ plane curve if and only if $$\int_{0}^{2\pi}\displaystyle\frac{cos \;\theta}{k(\theta)} d\theta=\int_{0}^{2\pi}\displaystyle\frac{sin\; \theta}{k(\theta)} d\theta =0$$
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 $\int_{0}^{L} Tds=0$. In the other direction, given an arbitrary $k$, the associated curve, up to translation is defined by $$x(\theta)=\int_{0}^{\theta}\displaystyle\frac{cos\;\eta \;}{k(\eta)}d\eta\;\;\;\;\;\;\;\;y(\theta)=\int_{0}^{\theta}\displaystyle\frac{sin\;\eta \;}{k(\eta)}d\eta$$ It is easy to check that this curve is closed and simple $(x(0) = y(0) = x(2\pi) = y(2\pi) = 0)$.