This might be silly, but I am not sure.
Let $A \subseteq \mathbb R^2$. Suppose that for any two points $x,y \in A$, I "add" the straight segment $[x,y]$ between them. Is the result convex?
That is, is $\cup_{(x,y)\in A^2} [x,y]$ convex?
2026-03-25 16:09:12.1774454952
Does connecting any two points in a set result in a convex set?
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No. If you take three non-collinear points then the union of the lines joining them is the triangle with those points as vertices. This is not convex.