# Minkowski's convex body theorem and binary quadratic forms

Saw this question in NZM and have a lot of difficulties trying to start.

Consider the binary quadratic form $f(x,y) = ax^2 + bxy + cy^2$ with $a > 0$ and $d = b^2 − 4ac < 0$. Show that there exist integers $x$ and $y$, not both zero, such that $\lvert ax^2 + bxy + cy^2\rvert \leq \frac{2}{\pi{}}\sqrt{−d}$.

I tried to use Minkowski's convex body theorem, but to no avail. What could be the possible convex symmetric subset of $\mathbb{R}^2$ to choose in this case such that the volume is greater than 4?

First note that the absolute value signs are superfluous because the assumptions that $d < 0$ and $a > 0$ imply that $f(x, y) \ge 0$. (See Definition 3.5 and Theorem 3.11. Those without the book can refer to the Wikipedia article on binary quadratic forms.)
The assumption $d < 0$ also implies that $$ax^2 + bxy + cy^2 \leq \frac{2}{\pi}\sqrt{−d}$$ represents an ellipse and its interior. That answers your question except for the following subtlety: