Does Minimum of Complex Set Subset Exist?

[Bashyboy]

Homework Statement

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 $L_B := \{|b|: b \in B \}$.

Homework Equations

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.

 

[micromass]

Think about an open set in $\mathbb{R}$.

 

[Bashyboy]

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?

 

[micromass]

Does a compact set always have a minimum?

 

[HallsofIvy]

A compact set is both bounded and closed.

 

[STEMucator]

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