With reference to Using a large set of elevation values for an area how can I find how 'hilly' it is?
which is a reasonable question, perfectly intuitive.
Clearly a 2 km radius near the fjords in Norway is more "hilly" than a similar radius around Topeka, Kansas.
So what is that intuition?
It's not the number of local maxima. Local variations produce local maxima and minima to an infinitely small scale.
It's not about global maxima or minima. Any finite set of elevations could have arbitrarily big or small variations.
But anyone can see that some places are more "hilly" than others.
The obvious answer is to add up the difference between the average elevation and the max or min. But that's not right because it doesn't take into account the locality of the max or min.
A better answer integrates slopes around every max and min, and this is a nontrivial formalization.
So my question is this: Is there a way to take a function of two variables, f(x, y), defined on a closed and bounded set, and average the gradient around every local max and min?