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Convex Set

  1. Aug 27, 2015 #1
    1. The problem statement, all variables and given/known data
    The following doesn't come from a textbook, and I am very uncertain whether it is true or false. Suppose that ##B \subseteq \mathbb{C}## is a convex set, and consider the set ##L_B := \{|b|: b \in B \}##.

    2. Relevant equations


    3. The attempt at a solution
    My question is, will ##min~L_B## exist? My thought was that ##B## being convex implied that ##L_B## is convex; but I am unsure whether convexity of ##L_B## is sufficient to conclude that ##min~L_B##. Please refrain from giving me an entire answer, but I would appreciate a few hints.
     
  2. jcsd
  3. Aug 27, 2015 #2

    micromass

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    Think about an open set in ##\mathbb{R}##.
     
  4. Aug 27, 2015 #3
    Ah, a counterexample! For instance, if we have ##B = (0,1)##, then ##L_B = B##, yet ##B## does not have a minimum. What if we stipulate that ##B## must also be compact?
     
  5. Aug 27, 2015 #4

    micromass

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    Does a compact set always have a minimum?
     
  6. Aug 28, 2015 #5

    HallsofIvy

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    A compact set is both bounded and closed.
     
  7. Aug 28, 2015 #6

    Zondrina

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    Since a compact set is bounded and closed, the infimum is the minimum, and the minimum exists.
     
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