On a direct and inverse problem in Potential Theory

The Newtonian potential of a homogeneous body D is proportional, outside D, to the Newtonian potential of a mass concentrated at a point x_0 in D if and only if D is a Euclidean ball centered at x_0. The if part simply follows from Gauss Mean Value Theorem for harmonic functions. The only if part is a Theorem by Aharonov, Schiffer and Zalcman. Aim of this talk is to present an extension of the previous result to non-homogeneous bodies, obtained in collaboration with Giovanni Cupini.