Pretty much that.
Does the fact that a set is convex mean that there will always be a closed form expression for its boundary? A closed form expression is once that can be evaluated in a finite number of operations.
Basically, I am wondering if the convexity of the set implies some sort of regularity to the boundary.
No. There are only countably many reals with closed form expressions. Take a triangle with all the corner coordinates not having closed form expressions, even if you are allowed to use the other coordinates. The sides will not have closed form expressions.