I believe that a polygon with a 180 degree angle would still be considered convex. In general, a convex region is defined as a region where the line segment connecting any two points in he region falls entirely within the region. In any case, the Happy End problem was originally considering a set of points in the plane in "general position", which meant that no three points were co-linear. In such a case, there could be no 180 degree angles.

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