A group presentation is "almost convex" if there is a constant C so that every pair of elements that lie distance r from the identity and distance at most 2 from each other in the word metric can be connected by a path that stays within r of the identity and has length at most C. Almost convexity is a nice property for groups to enjoy, since it implies a solvable word problem. However, the property depends on the choice of generating set, so is not a quasi-isometry invariant. In this talk I will discuss a generalization called "minimal almost convexity", which appears significantly weaker to begin with, yet it too implies a solvable word problem. A consideration of minimal almost convexity for the class of solvable Baumslag-Solitar groups yields the result that the property is not a quasi-isometry invariant. This is joint work with Susan Hermiller.