A convex polygon has no inlets or bays. Formally, whenever two points are contained in a comvex shape, the entire segment connecting those two points is contained in that shape. When the shape is a polygon, there is an equivalent definition. The polygon is convex iff every interior angle ≤ 180°. Let's prove this in both directions.
A larger angle implies an inward bend. Place a point on either side and draw a segment that cuts across the bay, hence part of the segment is outside the shape. Conversely, let the segment from p to q prove that a polygon is not convex. Let x be the point where the polygon first intersects the segment pq, and let y be the next point of intersection. Thus the segment xy is outside the polygon. Trace the polygon as it moves from x to y and let v be a point on this path that is farthest from the line xy. If v is not a vertex then it is part of a line segment that is parallel to xy. Move along this segment until you find an end point and call it v. Now v is a vertex whose adjacent sides form an angle that exceeds 180°.
