We extend the study of Chebyshev centers in pre-Hilbert C *-modules by considering the C *-algebra valued map defined by |x| = x, x 1/2. We prove that if T is a remotal subset of a pre-Hilbert C *-module M , and F ⊆ M is star-shaped at a relative Chebyshev center c of T with respect to F , then |x − q T (x)| 2 ≥ |x − c| 2 + |c − q T (c)| 2 (x ∈ F). The uniqueness of Chebyshev center follows from this inequality. This is a generalization of a well-known result on Hilbert spaces.