Existence of a minimer when dealing with a coercive functional: how to prove it?

Question

Let $H$ be an Hilbert space and let $C\subset H$ be a closed, convex and unbounded subset. Let $J:C\to\mathbb{R}$ be a functional.

Could someone helpe me to justify this sentence?

"If $J$ is coercive and $J(x_1)<+\infty$ forsome $x_1\in C$, then any minimizer of $J$ (if exists) on $C$ must occur for x inside some closed ball of radius $r>0$."

I guess it is related to the theorem of existence of a minimizer in Hilbert spaces, but I don't know how to prove that result holds true.

Could someone please help?

Thank you in advance!

Answer 1

This is just the direct application of the coercive-ness of $J$.

Since $J(x_1) <\infty$, any minimizer $y$ of $J$ must have $J(y) \le J(x_1)$. $J$ is coercive, so there is $r >0$ so that $J(z)> J(x_1)$ for all $z\in C$ with $|z| \ge r$. So any minimizer, if exists, must lie in the ball $\{y\in C : |y|<r\}$.