We apply a notion of geodesics of plurisubharmonic functions to interpolation of compact subsets of C n. Namely, two non-pluripolar, polynomially closed, compact subsets of C n are interpolated as level sets L t = {z : u t (z) = −1} for the geodesic u t between their relative extremal functions with respect to any ambient bounded domain. The sets L t are described in terms of certain holomorphic hulls. In the toric case, it is shown that the relative Monge-Ampère capacities of L t satisfy a dual Brunn-Minkowski inequality.