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Question: Assume you have a metric space $(E,d)$ and $A\subset E$ that is nonempty. Define $d_A:x \in E \rightarrow d(x,A)$. Show that $d_A$ is lipschitz and compute its Lipschitz seminorm.

My Question: My question is whether or not it is true that if you take $x,y,z \in E$ then $d_A(d(x,y),d(z,y)) \leq d(x,z)$.

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$d_A$ is a function from $E$ to the real numbers, right? But in your question you seem to have $d_A$ as a function of two real variables. – Gerry Myerson Oct 25 '12 at 2:20

Let $d_A(x) = \inf_{a\in A} d(x,a)$.

Now consider the triangle inequality: $d(x,z) \leq d(x,y)+d(y,z)$. Now suppose $z \in A$ and note that $d_A(x) = \inf_{a\in A} d(x,a)$. This gives $$d_A(x) \leq d(x,y)+d(y,z), \ \ \ \forall z \in A.$$ Now take the $\inf$ of the right hand side over $z \in A$. This gives you one side of the inequality. You should be able to finish from here.

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