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 $b$



crookedness of convex curves (milnor)
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.


