The stated assertion that a point outside some intermediate MES is on the boundary of the final MES seems unsupportable. It is true only for the first point removed from the point cloud and the next-to-final MES. How does this affect the correctness of the algorithm? Counterexample(?): 8 points of two tetrahedrons with non-overlapping circumspheres as point cloud, and perchance random sorting of points so that the vertices of the first tetrahedron are first and the vertices on the second tetrahedron last. Then the intermediate MES after last 4 encountered vertices is the circumsphere of the second tetrahedron, and the 4 points found outside that MES, which purportedly support the final MES, form the circumsphere of the first tetrahedron only.

The stated assertion that a point outside some intermediate Minimum Enclosing Sphere is on the boundary of the final Minimum Enclosing Sphere seems unsupportable. It is true only for the first point removed from the point cloud and the next-to-final MES.

How does this affect the correctness of the algorithm?

Counterexample(?): 8 points of two tetrahedrons with non-overlapping circumspheres as point cloud, and perchance random sorting of points so that the vertices of the first tetrahedron are first and the vertices on the second tetrahedron last. Then the intermediate MES after last 4 encountered vertices is the circumsphere of the second tetrahedron, and the 4 points found outside that MES, which purportedly support the final MES, form the circumsphere of the first tetrahedron only.