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Jul 3, 2016 at 14:04 comment added EM90 Thank you very much, Misha! Your argument on connectedness is really interesting. Actually, my definition of the set $U(x,y)$ is somehow intended to get a kind of "projection" or "shadow" of a cone type on the boundary, thus it is indeed related to that notion.
Jul 3, 2016 at 14:01 vote accept EM90
Jul 1, 2016 at 17:31 history answered Misha CC BY-SA 3.0