If you have two convex 2D shapes edit: one entirely inside the other, why must the smaller have a smaller circumference?
Last time I saw this mentioned, I don't know why I didn't see the solution. Now, the answer suddenly seems to jump out at me. Take a section of the inner shape length delta. Draw normals to the line at either end of it. Then the outer shape must cross between these lines, and this section of its curve must be at least as long as a straight line approximating the inner curve.
Then integrate. The sum of the inner lines tends to the inner circumference, the outer lines are at least as long.
(Technical faff. You need to assume all this converges, which it should because its a convex shape. It's a slight cheat because if the inner line is curved, it's slightly longer than the straight line we're comparing the outer to -- I *think* this could be fixed by observing that being curved makes the nromals bend apart. It would need to be to show the outer is *strictly* greater. Corners are special cases, make sure the partition of the circumference includes them, and take normals to the lines on either side, thus missing out a bit of the outer shape, which would need to be remembered to show strictly greater.)
I think the same thing should work fine for the 3d case, except you'd need a surface integral, which is a bit of a faff to actually calculate.
OK, and I use the maths tag for real for the first time.
Last time I saw this mentioned, I don't know why I didn't see the solution. Now, the answer suddenly seems to jump out at me. Take a section of the inner shape length delta. Draw normals to the line at either end of it. Then the outer shape must cross between these lines, and this section of its curve must be at least as long as a straight line approximating the inner curve.
Then integrate. The sum of the inner lines tends to the inner circumference, the outer lines are at least as long.
(Technical faff. You need to assume all this converges, which it should because its a convex shape. It's a slight cheat because if the inner line is curved, it's slightly longer than the straight line we're comparing the outer to -- I *think* this could be fixed by observing that being curved makes the nromals bend apart. It would need to be to show the outer is *strictly* greater. Corners are special cases, make sure the partition of the circumference includes them, and take normals to the lines on either side, thus missing out a bit of the outer shape, which would need to be remembered to show strictly greater.)
I think the same thing should work fine for the 3d case, except you'd need a surface integral, which is a bit of a faff to actually calculate.
OK, and I use the maths tag for real for the first time.