Consider the following world $($shown from an arial top$-$down view$)$:

a$)$ Consider discretizing this world $($i$.$e$.$ laying a grid down on top of the world so we see a discrete coordinate system$).$ Assume that the discretization is a fine as required $($but not continuous$)$ to preserve the geometry of the shapes present in the world. A convex shape is a shape where if we were to draw a line between any two points of the shape, the line would entirely be contained within the shape. Given a world that contains convex obstacles, is it possible for a vanilla hill climbing agent to get stuck?

b$)$ What if we could prove that the objective surface was convex. Would it be possible for a vanilla hill climbing agent to get stuck?