A Characterization of the Disc by the Angle of the Support Cone

Let K be a body in the plane and let λ < 1 be a positive real number. Suppose that for every x ∈ R 2 such that λ K + x is exteriorly tangent to K , the common support cone of K and λ K + x has a constant angle α . Suppose that one of the following conditions is satisfied: (a) K is a body of constant width, (b) K is centrally symmetric. Then K is a Euclidean disc.