In this paper, we introduce the \emph{uniform convex-like} property,
which is a new method of scaling the convexity of subsets of Banach spaces.
All bounded closed convex subsets of uniformly convex Banach spaces
and all compact convex subsets of strictly convex
Banach spaces are uniform convex-like.
Using this concept,
we prove a fixed point theorem in a Banach space.
We also show the existence of ergodic retractions
for nonexpansive mappings in a Banach space.