Let the radius of each outer circle be $a$, and let the radius of the inner circle be $r$.
The centers of the four outer circles form a square with side length $2a$.
The distance between the centers of two diagonally opposite outer circles is the length of the diagonal of the square, which is $2a\sqrt{2}$.
The distance between the centers of two diagonally opposite circles can also be expressed as the sum of the radii of the two circles plus twice the radius of the inner circle.
Therefore, we have $a + r + r + a = 2a + 2r = 2a\sqrt{2}$.
Dividing by 2, we get $a + r = a\sqrt{2}$. Solving for $r$, we have $r = a\sqrt{2} - a = a(\sqrt{2} - 1)$.
The radius of the inner circle is $a(\sqrt{2} - 1)$.
