Existence question: Can there exist a convex polygon witch is not a simple polygon?
For a polygon to be convex the outside angle of the polygon has to be more than or equal to 180 degrees. Now at intersection of 2 lines the outermost angle has to be less than 180 degrees for the lines to intersect between the endpoints.
Now the answer to this question depends on how you define what is inside of the polygon. If you consider some a even odd filling rule then the answer is No such shape exists. But if you consider only the outermost curve to do filling then yes, its possible if you intersect at the endpoint of a line. But to me this sounds more like splitting hairs as that shape reduces to a simple polygon.