Equivariant embeddings of metrizable proper G-spaces

For a locally compact group G   we consider the class G-MG-M of all proper (in the sense of R. Palais) G-spaces that are metrizable by a G  -invariant metric. We show that each X∈G-MX∈G-M admits a compatible G-invariant metric whose closed unit balls are small subsets of X. This is a key result to prove that X admits a closed equivariant embedding into an invariant convex subset V of a Banach G-space L   such that L∖{0}∈G-ML∖{0}∈G-M and V is a G  -absolute extensor for the class G-MG-M. On this way we establish two equivariant embedding results for proper G-spaces which may be considered as equivariant versions of the well-known Kuratowski–Wojdyslawski theorem and Arens–Eells theorem, respectively.