Suppose $(V, (\cdot, \cdot))$ is a Hilbert space, $U$ is a nonempty closed convex subset of $V$, and $g \in U$ is the unique element of $U$ with smallest norm. Prove that $\Re(g, h)\ge \|g\|^2 \ \forall h\in U$.
I was thinking about considering $f=\frac{1}{2}(g+h)\in U$. So $\|f\|^2 \ge \|g\|^2$. Unable to get the desired inequality. Any help?
As you noticed, for $t \in [0,1]$, using the convexity of $U$, $$ \|tg + (1 - t)h\|^2 \geq \|g\|^2 $$ Which, when expanding the inner product and dividing by $(1 - t)$ gives $$ 2t\Re(g, h) \geq (1 +t)\|g\|^2 - (1 - t)\|h\|^2 $$ Letting $t = 1$ gives the result.
Edit
As pointed out in the comments, 'letting $t = 1$' is imprecise and should be replaced with 'let $t \uparrow 1$'.