A polycrystal is a mixture of anisotropic materials (crystals) where each material may participate in a composite in any orientation. The effective conductivity tensor of such a composite depends on the microstructure of the composite. The set of effective properties one can obtain by mixing the same set of materials in different ways is called the G-closure of the original materials. The G-closure set has two important qualities: SO(3) invariance and a certain convexity property. In order to understand the interplay between these two properties we would like to understand SO(3) invariant functions with the convexity property. The first such result is due to Chandler Davis. In our case we examine what happens when the group action in Davis’s theorem is non-linear. In the process we uncover a simple abstract mechanism behind the Davis’s classical theorem. Our generalization features arbitrary groups, non-linear group actions and infinite dimensional vector spaces. We also gain extra flexibility to prove convexity of some G-invariant convex functions even though the theorem does not hold for all such functions. Even in the case of linear group actions on finite dimensional spaces we achieve a new generalization of Davis’s result.