  In this paper, we shall discuss topics in geometry of numbers in the positive
characteristic setting, such as covering radii. We find a closed form for
covering radii with respect to convex bodies, which will lead to a proof of the
positive characteristic analogue of Woods' conjecture in this setting. Then, we
will prove a positive characteristic analogue of Minkowski's conjecture about
the multiplicative covering radius. To do this, we shall prove a positive
characteristic analogue of Solan's result that every diagonal orbit intersects
the set of well rounded lattices. This implies that the Gruber-Mordell spectrum
in positive characteristic is trivial in every dimension.
