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Seems like you can find them by looking at the convex hull of your shape and making your evaluations only on the hul points.

Seems like you can find them by looking at the convex hull of your shape and making your evaluations only on the hul points.hull points.
If you want axis aligned "calipers" you may use cv::boundingRect().
If you want the smallest bounding rect, minAreaRect() might be helpful.