It's not clear to me why a geometrical proof is so hard to find. The definition says a polygon is convex if we can connect any pair of two points of the polygon with a line that's contained in that polygon. In most sources, for unknown reason, it is treated as an obvious fact.
This is the proof I've found - page 2. Could you verify whether it is correct? I have doubts with regard to this proof - there's a contradiction if we allow angles $> 180$, but how do we know if we can connect two points inside the polygon with a line inside the polygon, if it has angles $<180$? Maybe we would arrive at contradiction if we assumed angles $>170$. We should prove that all angles $< 180$ are okay here.