This is one of many observations from Pete L. Clark's questions on "Euclidean" quadratic forms. I sent Pete many positive integral forms that obeyed his condition. In turn, his condition turns out to be what Conway, Sloane, and in particular Gabriele Nebe refer to as "covering radius less than $\sqrt 2,$" see 

http://www.math.rwth-aachen.de/~nebe/pl.html 

and

http://www.math.rwth-aachen.de/~nebe/papers/CR.pdf   

One of Pete's relevant questions is

https://mathoverflow.net/questions/39510/must-a-ring-which-admits-a-euclidean-quadratic-form-be-euclidean


My own observations, confirmed by Pete with the Magma command CoveringRadius, are that the square of the covering radius is rational, and the denominator can always be taken to be a small power of 2 times the determinant of the lattice. Also the power of 2 seems to depend merely on the dimension.

So, that is the question, is the squared covering radius always rational with denominator a (dimension-dependent) power of 2 times the determinant of the lattice?

(note that it may be necessary to have the fraction not be in lowest terms to see the denominator as requested.)

(Also, Pete considers indefinite forms, other rings, etc. The covering radius stuff is for positive forms over the rational integers, which, except for the occasional annoying power of 2, are often called lattices. Not my fault).