An isodiametric problem with additional constraints

Let Mθ be the class of compact convex domains on the plane, which have diameter one, contain the origin, and are included in the fan region spanned by two rays starting from the origin with angle θ∈(π/3,π) between them. In this paper, we show that there is a unique compact convex domain with maximal area and also we determine the shape of the above domain.