Now you're reminding me of a wacky math conversation I had at Mathcamp [1] with a much smarter guy, who was talking about more esoteric definitions of volume in

Now you're reminding me of a wacky math conversation I had at Mathcamp [1] with a much smarter guy, who was talking about more esoteric definitions of volume in euclidean space. Something like: - n-dimensional volume is a function from (some) subsets of space to real numbers - it should be additive under union - it should scale by t^n when you scale the space by a factor of t I think the upshot of the conversation was that 0-dimensional volume of a shape should be its Euler characteristic. In the simple case of a finite set of points, the "volume" would be the number of points. And by your earlier comment, span({}) consists of a single point, so its volume should be 1. It all works! [1] https://www.mathcamp.org/ https://www.mathcamp.org/