Is there a notion of 'pre-convex' sets? A set $A$ in $\mathbb{R}^n$ is convex if for any two points $p,q \in A$ and real $\lambda \in [0,1]$, the point $\lambda p + (1-\lambda)q$ is also in $A$. There are many beautiful theorems about convex sets, perhaps the most important of which is: Let $f$ be a linear function on $\mathbb{R}^n$ (or more generally, take $f$ convex) and $A$ a compact, convex set. Then $f$ attains its maximum on the boundary of $A$. 
Another nice property is that any local maximum of $f$ (still linear/convex) restricted to $A$ is in fact a global maximum.
What I would like to know is if there is a notion for non-convex sets which still satisfy this local maximum property.
For example, the sphere is not convex, but clearly any local maximum of a linear function is a global maximum. Of course, the sphere is the boundary of the ball, which is convex.
My hunch is that if a set $N$ is the boundary of its convex hull then any local max is in fact a global max, but there seem to be weaker sets which also have this property. All the obvious examples to me (such as the 1-skeleton of a convex polyhedra, the sphere) are what I would want to call 'pre-convex,' but I am not sure exactly what that should mean.

Your conjecture is false. Define the function $f$ on the unit circle by
$$
   f(x,y) = x^2 + 3 y^2
$$
The unit circle is the boundary of a convex region given by $ x^2+y^2 \le 1$, you can chect that on the unit circle $f$ has a local maximum for $x=1, y=0$, but that is not a global maximum (the global maximum has the value 3).
