I just found out from my calculations the following:
Corresponding a given length of a one-dimensional element, if a two-dimensional lamina, (having same boundary length as the length of the one-dimensional element) is constructed, then a circle has the largest area.
Corresponding a given area of a two-dimensional lamina, if a three-dimensional structure,(having same surface-area as the area of the two-dimensional lamina) is constructed, then a sphere has the largest volume.
Now let us call length, width, height as "parameters" of the first, second, and third dimensions.
Then, I state:
- Any 1-D element has only one parameter-length.
- Any 2-D lamina has two parameters-length, width. Product of these gives Area of lamina.
Any 3-D structure has three parameters-length, width, height. Product of these gives Volume of structure.
Thus, mathematically, when constructing an "n"-dimensional entity, from an "n-1"-dimensional entity, the product of all n-dimensions is the greatest for that entity, that consists all of its "boundary-points", in n-dimensional space, equidistant from the origin of the n-dimensional coordinates-system.
I guess that this (the last statement) could be proved (or rather, disproved) using principles of mathematical-induction, but that's just a mere guess.