Let $\gamma$ be a smooth planar curve. Assume that $\gamma$ divides the plane into two domains and, it addition, that one of these domains is unbounded and convex. What can be said about the behavior of the curvature of $\gamma$ at infinity: does it tend to zero, if yes, is there a more precise asymptotics which holds in general?



Unbounded convex domains in 2D
No, the curvature does not need to go to $0$. Consider starting with a polygonal curve with infinitely many corners (e.g. the line segments joining $[n,n^2]$ to $[n+1, (n+1)^2]$ for integers $n$), and smoothing out the corners. If the angle at the $x=n$ corner changes by $\epsilon\_n$ and the piece of curve where this change occurs has length $\delta\_n$, the maximum curvature in this interval will be at least $\epsilon\_n/\delta\_n$. Thus if $\epsilon\_n/\delta\_n$ does not go to $0$, neither will the curvature.


