Distance from a sphere to a disjoint surface ($\mathrm{dist}(r{S}^{n-1},M)={\inf }_{x\in M}\Vert x\Vert -r$)

If a surface lies outside a sphere centered at the origin, its minimum distance to the sphere is its minimum radial distance minus the sphere radius. For each surface point, the triangle inequality gives that lower bound, and the radial point on the sphere attains it. Thus minimizing the squared norm on the surface suffices.