I'm currently reading On the Total Curvature of Knots and am trying to understand one of the lemmas in it (3.3)
A closed polygon $P$ in $H^2$ is convex if and only if for every $b$ either $\mu(P,b) = 1$ or $\mu(P,b) = \infty$
I understand the argument for why you need $\mu(P,b) = 1$, but the $\infty$ part confuses me:
I fail to see how a polygon with an infinite number of maxima can be considered convex.
Any help would be apreciated,
$\mu(P,b)$ denotes the number of local maxima on the height function, i.e. the projection of the polygon onto some unit vector $b$