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This post records a variant to the question asked in this post: Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon

  • Given a convex n-gon, how does one algorithmically find the interior point that minimizes the maximum angle subtended at the point by an edge of the polygon? Is the optimal point always unique?

Remark 1: It is easy to see that the answers to the present question and the above linked one of trying to maximize the least angle subtended could be different. For example, consider a regular n-gon - for which both questions have the centroid as the optimal point - and make it a convex n+1-gon by adding a vertex close to one of the n pre-existing vertices. Now, for sufficiently large n, the interior point minimizing the max subtended angle remains at the centroid but the point maximizing the smallest subtended angle moves towards the newly added short edge.

Remark 2: If the convex point is a triangle, the interior point that minimizes the largest angle subtended there by an edge is unique and is the Steiner point.

Remark 3: Further variants that seek interior point(s) minimizing the range of angles subtended there by the edges or the standard deviation among the angles subtended there by the edges could also be considered.

Nandakumar R
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  • I’d say that, in the notation of the linked post, the min max angle is the maximum $\theta$ such that the $n$ disk segments ${S(e,\theta)}{e\in\mathcal E}$ covers the polygon $C$ (denoting here $\mathcal E$ the set of the edges), while the maxmin angle was the maximum $\theta$ such that the $n$ disk segments ${S(e,\theta)}{e\in\mathcal E}$ have nonempty intersection. This I think implies the uniqueness of the minmax point by an argument analogous to that for the maxmin point. – Pietro Majer Mar 28 '24 at 21:31
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    Actually this argument only gives a local uniqueness (hence finiteness). In fact in a rectangle say 1x1000 certainly the center (which is the maxmin point) is certainly not the minmax point, so by simmetry there are (at least) two maxmin points – Pietro Majer Mar 29 '24 at 06:14
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    Here’s a conjecture: the max number of max-min angle points for an $n$-gons is $n-2$ – Pietro Majer Mar 29 '24 at 07:29
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    Guess the basic difference between this question and the earlier one is that here one has to handle intersections of non-convex regions - this might also be the reason why there are multiple optimal points. – Nandakumar R Mar 29 '24 at 13:30

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