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Question: In $\mathbb{R}^{3}$, Can we find a line L and a closed convex set S with $S \cap L = \varnothing$ such that for each plane $\Pi$, $L \subseteq \Pi$ we have $\Pi \cap S \neq \emptyset$?

How can we find these line $L$ and the closed convex set $S$?

TrItOs
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  • Where does this question come from, please? – Gerry Myerson Feb 01 '24 at 23:36
  • (And how does this look, one dimension down? In the plane, can we find a point $P$ and a closed convex set $S$ with $P$ not in $S$ but with every line containing $P$ meeting $S$?) – Gerry Myerson Feb 01 '24 at 23:38
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    @GerryMyerson: This Question is coming from the textbook "Steven R. Lay - Convex Sets and Their Applications" Exercise 4.11 – TrItOs Feb 02 '24 at 14:31
  • @GerryMyerson: One dimension down it is not true! Because there is a line containing P but not intersects S – TrItOs Feb 02 '24 at 14:34
  • OK, so what happens if you project the line down to a point, so each plane containing the line projects to a line containing the point. Do closed convex sets in 3 dimensions project to closed convex sets in 2 dimensions? Could you then apply the one-dimension-down result? – Gerry Myerson Feb 02 '24 at 21:11
  • Your question asks, "Can we find a line ....?" but what I can see of Lay's book on google says "Find a line ...." – Gerry Myerson Feb 02 '24 at 21:19
  • @GerryMyerson: Okay "Find a line L and a closed convex set S..." Can you help me with the solution? – TrItOs Feb 04 '24 at 15:13

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