Extending equivariant maps into spaces with convex structure

We prove that if G is a compact Lie group, Y a G-space equipped with a topological local convex structure compatible with the action of G, then Y is a G-ANE for metrizable G-spaces. If, in addition, Y has a G-fixed point and admits a global convex structure compatible with the action of G, then Y is a G-AE. This is applied to show that certain hyperspaces related to the Banach–Mazur compacta are equivariant absolute extensors.