Good and bad radii of convex polygons

The 'radii' considered here are the inradius ρ, the circumradius R, the diameter δ, and the width Δ. The convex polygons in question have their vertices at points of the integer lattice in R², and their radii are measured with respect to an ℓ^p norm. Computation of these radii for convex polygons (and of their higher-dimensional analogues for convex polytopes) is of interest in connection with a number of applications, and may be regarded as a basic problem in computational geometry. The terms good radius and bad radius refer to the existence or nonexistence of a rationalizing polynomial - a nonconstant rational polynomial q such that q(φ(C)) is rational whenever C is a convex lattice polygon and φ is the radius function in question. When a radius is good, the polynomial is a tool for implicit computation of the radius in the binary model of computation; otherwise it seems to be necessary to resort to approximation. It is proved here that all four radii are good when p member of {1, ∞}, while δ is good when p is an integer and Δ is good when p/(p - 1) is an integer. Thus δ and Δ are both good when p = 2, and it turns out that R is also good in this case. However, the main results are that r is bad when p = 2 and R is bad for each integer p ≥ 3.