  For functions defined on C^n or (R_+)^n we construct a dequantization
transform, which is closely related to the Maslov dequantization. The
subdifferential at the origin of a dequantized polynomial coincides with its
Newton polytope. For the semiring of polynomials with nonnegative coefficients,
the dequantization transform is a homomorphism of this semiring to the
idempotent semiring of convex polytopes with the well-known Minkowski
operations. Using the dequantization transform we generalize these results to a
wide class of functions and convex sets.
