Is set A convex when set A = the union of all points on a circle and all points in the interior of the circle, minus point R?
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It depends on where $R$ is. – Nov 07 '18 at 01:47
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You are asking if a punctured disc, similar to an annulus is convex? So long as the point where the puncture occurs is in the interior, then no. Suppose the puncture occurred at point $(x_0,y_0)$. Then for sufficiently small $\epsilon$ you have $(x_0-\epsilon,y_0)$ and $(x_0+\epsilon,y_0)$ do not have the entire line segment connecting them in your set. If it is on the exterior, then sure it is. – JMoravitz Nov 07 '18 at 01:47
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Sorry! Forgot to add. Point R is on the circle. – Jolly Nov 07 '18 at 02:23