Let $\cal H$ is a finite dimension Hilbert space, $G$ is a convex functional, i.e. $\forall x, y \in \cal H$, $G(a\cdot x+(1-a)\cdot y) \le aG(x) + (1-a)G(y), \forall a \in [0,1]$. Let $S \subset H$ ($S$ is not empty and $S \neq \cal H$) are minimizers of $G$ on $\cal H$, i.e. $\forall x, y \in S, G(x)=G(y)$ and $\forall z \in {\cal H}, z \notin S, G(z) > G(x)$. How can I approve that if $S$ is bounded then $G$ is coercive? i.e. if $\forall x \in S, \|x\| \le M$, and $\forall y \in \cal H$, $G(y) \rightarrow \infty$ if $\|y\| \rightarrow \infty$.
It will be great that I can get a simple proof for this.
It seems that if $\cal H$ has infinite dimension, the result is not correct. So, we add a requirement that $\cal H$ has finite dimension.