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I have the intuition that the minimal point (in the sense of having the lowest value in one of the Euclidean space dimensions) of a intersection of $n$ convex sets is always the minimal point of the intersection of two convex sets in the family. Has this been proven before?

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In $d$ dimensions, you need at least $d$ convex sets, not two. For example, in a suitably rotated polytope in $d$ dimensions, the minimal point will be a vertex, which is defined by the intersection of $d$ planes.

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Many thanks Geoffrey, my problem is in two dimensions. Is there a known result that states that in $d$ dimensions you need at least $d$ convex sets to define a minimal point? –  Adrian TC Oct 31 '12 at 17:32
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