In this paper we improve an earlier result by Bukhgeim and Uhlmann [1], by showing that in dimension n ≥ 3, the knowledge of the Cauchy data for the Schr¨dinger equation measured on possibly very small subsets of the o boundary determines uniquely the potential. We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem. This implies a similar result for the problem of Electrical Impedance Tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary. .
