I am trying to find out if there is any known result in convex optimization that implies the following statement:
"A minimal point of the intersection of $N > 2$ convex sets in $\mathbb{R}^2$ is a minimal point of the intersection of $2$ sets among the aforementioned $N$."
and more generally:
"A minimal point of the intersection of $N >d$ convex sets in $\mathbb{R}^d$ is a minimal point of the intersection of $d$ sets among the aforementioned $N$."
Note: Minimal point is considered here to be a point in the set that attains the lowest value in one of the Euclidean space dimensions.
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$\begingroup$ I don't understand your "Note"--it makes no sense to me. $\endgroup$– Włodzimierz HolsztyńskiFeb 12, 2013 at 4:38
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$\begingroup$ I'd like to dwell on your question, but I need a complete, precise definition of "minimal point", as you use it. $\endgroup$– Włodzimierz HolsztyńskiFeb 12, 2013 at 4:40
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1 Answer
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This seems to be an immediate consequence of Helly's theorem, applied to the given convex sets and the set of points where the space-dimension in the "Note" is smaller than its minimum over the intersection.
answered Dec 9, 2012 at 19:58