Reflexive Banach Space: For any $x ∈ E$, there exists $x_0 ∈ M$ so that $|x−x_0| =\inf\{|x−y| :y ∈ M \}$

Question

Let $E$ be a reflexive $ \mathbb K$−Banach space and assume that $M$ is a convex, closed, bounded subset of $E.$ Prove that for any $x ∈ E$, there exists $x_0 ∈ M$ so that $|x−x_0| = \inf\{|x−y| :y ∈ M \}$.

I got the hint to use Fenchel-Rockafellar Theorem (also known as Fenchel's Duality Theorem) and then use the weak* topology $\sigma(E^*,E)$.

But I don't see the use of applying this theorem in this case. Why does one need this and how does one prove this statement using this particular theorem?

Answer 1

It is the classical result in convex analysis. You do not actually need Fenchel duality here if you combine the following three basic facts:

• Any closed convex bounded set is weakly compact in a reflexive Banach space
• Norm is weakly lower semicontinuous
• Weierstrass extreme value theorem

P.S. The set does not have to be bounded if you use the third mentioned variant of the Weierstrass theorem.