Is there a finite set $\mathcal{D}$ such that $E(a, b, c) = \sum_{x \in \mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2$ is strongly convex?

Question

Let $\mathcal{D} \subset \mathbb{R}^2$ be a finite set. Define a function $E : \mathbb{R}^3 \rightarrow \mathbb{R}$ by $$\large E(a, b, c) = \sum_{x \in \mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$

Does there exist a set $\mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.

I proved $E$ is convex, and I know that if $\mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $\mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $\mathcal{D}$ has multiple elements are appreciated.

Answer 1

Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.