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Let $ S\subset\mathbb{R}^n$ , with $ n\geq 3$ , such that for any hyperplane $ L$ one has $ L\cap S$ closed, semialgebraic, and convex. Is it true that $ S$ itself is semialgebraic? A colleague explained to me that closedness is necessary, as one can take an open ball in $ \mathbb{R}^n$ and pasteRead more