# Affine Curvature

I was reading a paper related to convex curves, and encountered the following quantity:

Let $\gamma$ be a convex curve and $k(\cdot)$ be its curvature (with respect to arc length). The author is using the following highly ambiguous (at least to me) notation: $$\int_\gamma k(s)^\frac{1}{3} ds$$ and calling this quantity the affine curvature. My question is:

$\textbf{What does this notation mean?}$

First of all, what exactly is $k(s)?$ Secondly, what does it mean to integrate $k(s)$ with respect to $s$ along the curve $\gamma$? Any help will be greatly appreciated!

• $k(s)$ probably stands for the curvature as a function of the arc length $s$. Integrating over $ds$ means summing up all the quantity over the curve itself. Commented Nov 18, 2017 at 2:21
• Is there a rigorous definition of the integral?
– abcd
Commented Nov 18, 2017 at 2:25
• It's just the usual line integral of a scalar function... Commented Nov 18, 2017 at 4:52

As stated by me and Anthony Carapetis in the comments, $k(s)$ is just some scalar function and integrating it over $ds$ means that you are integrating along the curve. Hence, this is a line integral, more rigorously speaking.
Essentially what you are doing is summing up all cubic roots of the curvature $k(s)$ alongside the curve $s$.
Notice that to calculate the integral you would need a parametrization $s(t)$ and then integrate $k(s)$ wrt $t$.