I would be grateful if someone can check my claims or recommend another way of solution:
Let $n \in \mathbb{N}^{*}(n \neq 0)$, $A$ a real square matrix, and $\mathbf{c}$ a vector in $\mathbb{R}^{n}$. Consider a real function $h: \mathbb{R} \longrightarrow \mathbb{R}, g \in C^{2}(\mathbb{R})$, and we introduce the function $g: \mathbb{R}^{n} \longrightarrow \mathbb{R}$, defined by $$ g(\mathbf{x})=h\left(\|A \mathbf{x}\|^{2}\right)-<\mathbf{c}, \mathbf{x}>, \quad \forall \mathbf{x} \in \mathbb{R}^{n} . $$
I have managed to understand that $$\nabla g=h^{\prime}(\langle A x, A x\rangle) \cdot2 A^{\top} A x-c$$, $$H(g)=2 h^{\prime\prime}(\langle A x, A x\rangle) A^{\top} A$$
a. Assume that the matrix $\mathrm{A}$ is invertible $\left(A^{-1}\right.$ exists). Assume also that there exist $\alpha>0$ such that, for all $z \in \mathbf{R}$, $$ h^{\prime}(z) \geq \alpha, \quad h^{\prime \prime}(z) \geq 0 . $$ Prove that $f$ is an elliptic function.
b. Assume $h(z)=z, \forall z \in \mathbb{R}$, and that the matrix A is invertible. I want to prove that $g$ is elliptic and deduce that there exists one, and only one, minimum $\mathrm{x}^{\star}$ of $f$ on every closed, convex subset $U$ of $\mathbf{R}^{n}$.
My attempt
a. The definition of elliptic is $\exists c>0, \forall x, y \in \mathbb{R}^{n}:\langle\nabla g(x)-\nabla g(y), x-y\rangle \geq c\|x-y\|^{2} $. There is also another condition that is equivalent to elliptic:
iff $\exists \beta>0$ $ \langle H(g) \cdot x, x\rangle \geqslant \beta\|x\|^{2} \quad, \quad \forall x \in \mathbb{R}^{n} $
I thought using $$ \left\langle 2 h^{\prime \prime}(\langle A x, A x\rangle) A^{\top} A \boldsymbol{x}, \boldsymbol{x}\right\rangle \geqslant 2 h^{\prime \prime}(\langle A x, A x\rangle) \lambda_{\min }\|x\|^{2} $$
Now I know that the matrix in $ H(g)$ is symmetric but not positive semi-definite, so how to continue or change? why is it given that $A$ is invertible?
b. $g(\mathbf{x})=\|A \mathbf{x}\|^{2}-<\mathbf{c}, \mathbf{x}>$. Does it make a simpler case for ellipticity? Is there theorem to prove that there exists such a minimum?