Say you have a city with two skyscrapers on opposite sides of town, one 49 stories high and the other 51 stories high. When you're at the top of the 49-story building, you're at a local maximum of height and close to the global maximum, but you're not exactly a short gradient-ascent jaunt away from the global maximum.
Sure, but how much work are you going to put in to go up two floors? Local optimum + close to a global optimum means there's not much cost benefit in changing.
Not really, depends on definition of "close", i.e. how you measure distance. If closeness means distance in search space, then you're right, but it also could mean local vs global are close in value being optimized, but they are located far from each other in search space.