By using a family of harmonic functions introduced by Ulku Kuran in 1972, we define a new harmonic invariant that measures the gap between the perimeter of a domain D and the perimeter of the biggest ball contained in D and centered at a fixed point x_0 of D. From the properties of this harmonic invariant we get new proofs, generalizations and partial improvements of several rigidity and stability Theorems by Lewis and Vogel, by Preiss and Toro, by Fichera and by Aharonov, Schiffer and Zalcman. The complete proofs of all these new results will appear in a paper in collaboration with Giovanni Cupini.