Spheres packed in a spherical shell.

In general, join the centers of all four equal spheres of radius r to get a regular tetrahedron & then distance of vertex from the center of tetrahedron thus one can easily find the minimum radius R of spherical shell

$R_{min}=\text{distance of apex from center of tetrahedron }+\text{radius of small sphere}$

$R_{min}=r\sqrt{\frac{3}{2}}+r=r\left(\sqrt{\frac{3}{2}}+1\right)$

or minimum diameter $D_{min}=2R_{min}=2r\left(\sqrt{\frac{3}{2}}+1\right)=d\left(\sqrt{\frac{3}{2}}+1\right)$

where, d is the diameter of equal spheres,

now, setting the value of d=10 mm, minimum diameter $D_{min}=10\left(\sqrt{\frac{3}{2}}+1\right)\approx 22.24744871 mm$