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I am reading the paper (Ann Math. 1950) "on the total curvature of knots" by J. Milnor. I was trying to understand this part of the paper (here is a free access link to the paper):

Let $P$ be a polygon in $\mathbb{R}^{n}$ let $Q$ be its spherical image on the unit sphere. let $b$ be a point of $S^{n-1}$ and denote $S_{b}^{n-2}$ the great sphere of $S^{n-1}$ which has a pole at $b$. Let $b_{j}$ and $b_{j-1}$ be to consecutive vertices of $Q$, then Q crosses $S_{b}^{n-2}$ iff $b.(a_{j+1}-a_{j})$ and $b.(a_{j}-a_{j-1})$ have opposite signs, so that $b.a_{j}$ is a maximum or minimum on the $P$.

Let $\{a_{j}\}$ be the vertices of polygon and let $b_{j}=\frac{a_{j+1}-a_{j}}{|a_{j+1}-a_{j}|}$. And I think $S_{b}^{n-2}$ is the intersection of the hyperplane with normal vector $b$ and the unit sphere.

My question is that why should $b.a_{j}$ be maximum or minimum on $P$?

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It is local minimum or maximum --- at the point $a_i$ the derivative of $b⋅α$ changes sign before it has sign of $b⋅(a_j-a_{j-1})$ and after the sign of $b⋅(a_{j+1}-a_j)$.

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  • $\begingroup$ I made an answer from my comment to remove the question from "unanswered". $\endgroup$
    – Anton Petrunin
    Commented Dec 19, 2022 at 14:22

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