An Isodiametric Problem with Additional Constraints in Euclidean space R^3

Let $C_{\theta }$ be a circular cone in Euclidean space ${\mathbb{R}}^3$,which apex is the origin and apex angle of the cone is $\theta\in \left(\pi/3, \pi\right)$\mathrm{.}Let$M_\theta$ be the class of compact convex domains in Euclidean space ${\mathbb{R}}^3$, which have diameter one, contains the origin and are included in $C_{\theta }$. In this paper, we show that there is a unique compact convex domain with maximal volume and also we determine the shape of the above domain.