Let K be a number field of degree n with a fixed embedding in the complex numbers. Let | . | be the normalized absolute value given by that embedding. (The square of the ordinary absolute value if the embedding is non-real.) Does there exist a basis x_1 , ... , x_n for K over the rationals with the following property:
| r_1 x_1 | + ... + | r_n x_n | = the maximum of | r_1 x_1 + ... + r_n x_n |_v
where | . |_v runs through the normalized archimedean absolute values of K ?
Later: As posed, the answer is no. Suppose I started with the field generated by a cube root and was unlucky enough to start with the real embedding. Then I'd be in effect trying to show ( x + y + z )^2 > x^3 + y^3 + z^3 on the positive octant. But if I was lucky enough to start with the complex embedding ...