# convexity and ellipticity condition

Given $$z \in \mathbb{R}^n$$ and $$x \in \mathbb{R}$$, we are given the following map $$f: \mathbb{R}^{n+1} \rightarrow \mathbb{R}$$, and $$\alpha_{i,j}(x) \in L^{\infty}(\mathbb{R})$$ :

$$f(x,z) = \sum_{i,j=1}^{n} \alpha_{i,j}(x) z_i z_j$$

I would like to check the map $$z \mapsto f(x,z)$$ is convex (for almost every $$x \in \mathbb{R})$$ in the case the ellipticity condition is satisfied.

For $$z, w \in \Bbb R^n$$, $$0 < \lambda < 1$$, and $$1 \le i, j \le n$$ is:
$$(\lambda z_i + (1-\lambda) w_i)(\lambda z_j + (1-\lambda) w_j) - \lambda z_i z_j - (1-\lambda) w_i w_j \\ = -\lambda (1-\lambda) (z_i z_j - z_i w_j - z_j w_i + w_i w_j) \\ = -\lambda (1-\lambda) (z_i - w_i)(z_j - w_j) \, .$$
It follows that for every $$x$$, satisfying the ellipticity condition,
$$f(x, \lambda z + (1-\lambda)w) - \lambda f(x, z) - (1-\lambda) f(x, w) \\ = -\lambda (1-\lambda) \sum_{i,j=1}^{n} \alpha_{i,j}(x) (z_i - w_i)(z_j - w_j) \\ \le -\lambda (1-\lambda) \eta \Vert z-w \Vert^2$$
and that is strictly negative if $$z \ne w$$, so that $$z \mapsto f(x,z)$$ is strictly convex.