We improve the isoperimetric inequality on finitely generated groups. We calculate the isoperimetric profile for homogeneous trees, accompanied by examples of optimal subsets. Additionally, we provide guidelines for constructing new optimal subsets.

We generalize the notion of almost convexity on Cayley graphs to metric spaces, introducing the concept of metric almost-convexity. Furthermore, we demonstrate that horospherical products do not satisfy the Poénaru condition.

Finally, we prove that if two geodesics metric spaces are rough-isometrics and one of them satisfies the condition of metric almost-convexity, then the other space also satisfies this condition.