Covering of a Reduced Spherical Body by a Disk

We prove the following theorems: (1) every spherical convex body W of constant width $$ \varDelta (W)\ge \frac{\uppi}{2} can be covered by a disk radius (W)+\arcsin \left(\frac{2\sqrt{3}}{3}\cos \frac{\varDelta (W)}{2}\right)-\frac{\uppi}{2}; (2) reduced R thickness (R)<\frac{\uppi}{2} arctan \left(\sqrt{2}\tan (R)}{2}\right).