On topological groups with an approximate fixed point property

A topological group $G$ has the Approximate Fixed Point (AFP) property on a bounded convex subset $C$ of a locally convex space if every continuous affine action of $G$ on $C$ admits a net $(x_i)$, $x_i\in C$, such that $x_i-g{x}_i⟶0$ for all $g\in G$. We study the relationship of this property with amenability.