What is the largest radius of the circle?

Let $S$ be the set of pairs of real numbers $(x,y)$ such that for at least one pair of positive integers $m,n$ ,

$mn(2(xm+yn) - (x^2+y^2)) > m^3n+n^3m - \frac{1}{4}$

As it turns out, for some values of $r$ , given any closed circle $C_r$ of radius $r$ , $S\cap C_r$ is always connected.

What is the largest value for $2r$ that satisfies this condition? Give to three decimal places.

Note: The null set is (trivially) connected.