Question

hello,

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,

Thanks

Edit:

$\mu(P,b)$ denotes the number of local maxima on the height function, i.e. the projection of the polygon onto some unit vector $b$

Answer 1

If you have a polygon with say a horizontal side, each point is a maxmimum (or minimum) of the projection onto the $y$-axis. So we must admit the possibility of an infinite number of maxima.