91Ó°ÊÓ

In the mathematical study of curves, **curvature** is a measure of how sharply the curve bends at a given point. For a curve parameterized by arc length, denoted as \(\alpha(s)\), the curvature \(k(s)\) is defined as the magnitude of the derivative of the unit tangent vector with respect to the arc length parameter \(s\). This is given by the formula \(k(s) = | \frac{dT}{ds} |\), where \(T\) is the unit tangent vector.

For a regular closed curve, it is essential that the curvature \(k(s)\) be non-zero and finite for all points along the curve. This ensures the curve doesn't have any sharp cusps or discontinuities. In our exercise, we know that \(0 eq |k(s)| \leq 1\), indicating that the curvature is always positive up to a maximum value of 1.

The curvature plays a crucial role in understanding the geometry and shape of curves. By examining how curvature behaves along different segments, we can infer properties like smoothness and convexity of the curve.

**Fenchel's Theorem** is a fundamental result in differential geometry that provides a relationship between the curvature of a closed space curve and its total length. The theorem states that for any closed curve \(\alpha: [0, l] \rightarrow \mathbb{R}^3\) parameterized by arc length, the total curvature satisfies the inequality:

\[ \int_0^l |k(s)| \, ds \geq 2\pi \]

This inequality tells us that the total integral of the absolute value of curvature over the length of the curve must be at least \(2 \pi\). If this inequality holds with equality, meaning \(\int_0^l |k(s)| \, ds = 2\pi\), then the curve must be a plane convex curve. A plane convex curve lies entirely in a single plane and is convex, meaning it has no indentations (the line segment between any two points on the curve lies entirely within the curve).

In the given exercise, we utilize Fenchel's Theorem to show that the length \(l\) of the curve must be at least \(2 \pi\). By applying this theorem, we conclude that if \(l = 2 \pi\), then the curve must be a plane convex curve.

**Arc length parametrization** is a special way of parameterizing curves such that the parameter directly represents the distance along the curve. For a given curve \(\alpha(t)\), the arc length \(s\) from a starting point \(t_0\) is given by the integral:

\[ s(t) = \int_{t_0}^t | \alpha'(\tau) | \, d\tau \]

When parameterized by arc length, the curve is denoted as \(\alpha(s)\). In this parameterization, the speed of traversal along the curve is constant and equal to 1, i.e., \(\alpha' (s) = 1\).

In our exercise, the curve is already given in arc length parametrization, which simplifies our analysis. It allows us to directly apply Fenchel's Theorem without needing to reparametrize the curve. This simplification means we can focus solely on the properties of the curvature \(k(s)\) and the integral bounds it implies.