I come from a statistics background, so we often frame things in the context of distributions and the parameters that characterize those distributions. So the most common parameter is the average value of some distribution.
Now, I was wondering whether something like a torus could have an average value? I am thinking of the simple torus that resembles a donut with a hole. That is the question, can we establish a notion of an average value for something like a torus, the same way that we can estimate an average for a 3 dimensional cloud of points?
We can apply the standard statistical algorithm to find the average. The tricky thing is that if I were to project the 3-d torus into 3 separate 1-dimensional projections, take the average value of each projection, and then recombine those averages into a 3-tuple of values, I would essentially obtain a point that was inside of the donut hole, right.
So the problem then, is that in a definitional sense we are saying that the average or central tendency of random points taken from a torus would be a point that is not in the torus--but in the hole of the torus. So this is a point that could never be reached, but which is used as the central tendency.
Now I can appreciate that a point outside of a set can represent or characterize the tendency of the set--in the same way that in point-set topology the limit of a convergent sequence can lie outside of the open set. So that is fine.
But I was wondering if there are any other ideas within perhaps differential topology or even measure theory/functional analysis which find a way to reconcile this notion of an average value with that value needing to be reachable from within the set?