# 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 $$r\in\mathcal C^2([0,L],\mathbb R^2)$$,

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. I.e. the curve is the topological boundary of a convex set.

How can I prove that both definitions are equivalent?

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 $$x(0) = y(0) = x(2\pi) = y(2\pi) = 0$$.