I don't know the best way to phrase this (english isn't my main language), so I'll do my best.
First we have the perimeter (one dimension, a line), which is conserved after a deformation. for example if you have a square and deform it (by this I mean you alter its shape but essentially the line which conforms it, is the same, just in a different shape, you can maybe build a triangle using that same line or something) then the perimeter is the same.
In R^3 the volume of an object is also conserved after deformation. But in R^2 this doesn´t happen with the area. In the previous example of altering a square, it will keep its perimeter but not necessarily its area.
Because of this I had this question. Does this happen for all odd dimensions (R^(2n+1)), if so why?
I'll be happy to clarify anything. Thanks in advance!