Reduced bodies in normed planes

We say that a convex body R of a d-dimensional real normed linear space Md is reduced, if Δ(P) < Δ(R) for every convex body P ⊂ R different from R. The symbol Δ(C) stands here for the thickness (in the sense of the norm) of a convex body C ⊂ Md. We establish a number of properties of reduced bodies in M2. They are consequences of our basic Theorem which describes the situation when the width (in the sense of the norm) of a reduced body R ⊂ M2 is larger than Δ(R) for all directions strictly between two fixed directions and equals Δ(R) for these two directions.