Let $C$ and $C'$ be two convex subsets of $\mathbb{R}^n$ with nonempty intersection. Is it true that $$ \overline{C\cap C'} = \overline{C}\cap \overline{C'} ? $$
2026-03-29 21:58:23.1774821503
Closure of the intersection of two convex sets
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No. You can take $H_1 = \{(x,y) \in R^2, y > 0 \}$ and $H_2 = \{(x,y) | y < 0 \}$. Then let $p = (0,0)$, and let $A = H_1 \cup p$ and $B = H_2 \cup p$.