16 Nov
2020
16 Nov
'20
11:40 a.m.
On Nov 16, 2020, at 1:28 PM, James Propp <jamespropp@gmail.com> wrote:
The necessity of their having self-intersections in 3D is a consequence of the linearity of the transformation, but I don't see an instant proof of that at the moment.
Consider the corresponding statement for continuous curves. Suppose surface S is the Minkowski sum of two closed curves. Express the signed-volume of S as a double integral over its two parameters. When S is a sum of curves, and we use them as parameters, this integral is trivially 0x0 = 0. But if S can be immersed without self intersections it would have a nonzero volume. -Veit