On the inverse of Newton’s no-gravity-in-cavity theorem

Newton in his Principia (first book) claims and proves that spherical shells do not exert gravitational force inside the cavity of the shell. This remarkable statement, Newton proved using geometric methods, since calculus was yet not born, but understood by Newton. This result was later extended to ellipsoidal shells (homoeoid) by P.-S. Laplace, using computation, and soon after by J. Ivory using a more geometric approach.
In early 30’s P. Dive proved the inverse of this theorem, for ellipsoidal shells. Since then there have been several other proofs, using different approaches.
In this talk, I shall recall Ivory’s partially geometric proof and then extend this result to paraboloids, and generally to unbounded domains.
Our result in terms of global solutions of free boundary problems implies a complete classification in dimensions greater than five. In dimensions two this was done by Makoto Sakai, some 30 years ago.