  In an earlier work, the first author and Petsche solved an energy
minimization problem for local fields and used the result to obtain lower
bounds on the height of algebraic numbers all whose conjugates lie in various
local fields, such as totally real and totally p-adic numbers. In this paper,
we extend these techniques and solve the corresponding minimization programs
for real intervals and p-adic discs, obtaining several new lower bounds for the
height of algebraic numbers all of whose conjugates lie in such sets.
