PSA for Clive and Calamrin - Surface area of two nested convex shapes

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.

Wake (literally)

I naughtily skipped the start with a Ceilidh. There was hmmm_tea, Rachel C and Rachel C's Dave, atreic and emperor, ilanin, theinquisitor and (the supernaturally cuddly) rjw76, numberland (thanks for dancing advice :)), Joy (so I did see all my companions), and a couple of people I've seen about at other ceilidhs but forgot the name of.

An all night event is good to play games you'd never have time to. We spent ages with ptc24's Peurto Rico, so now I know how to play, and if I skim some strategy might even be able to do so acceptably with non-beginners at some point.

Fruit bandits is a very silly game that works quite well, though I'd be curious to see it with fewer people. Matthew and Matt pressed tomatoes on to me (metaphorically), making it an amazingly fruit-themed evening.

angoel's pirates game was very funny. Again, it needs fewer people or tweaking, but it's amazing how "pirate fighting sauron" goes on being funny.

beckyc dressed up, and was more gorgeous than anything I've seen in quite a while. I hope this is any consolation for asthma troubles :(

I dashed home for some supplies, slightly worrying people I might miss the Afmaelisdagr, and what would happen then?

The morning was clear, crisp, blue, and generally perfect. Castle mound was full of some other people watching the sunrise, and we thought "What are these strange people doing here?" though I'm sure not as much as *they* thought "What are these strange people doing here?"

I managed the ceremony and the new sun. The invocation was a bit different, but seemed to work at least as well: we wound the sun up a bit more, so it seemed to take longer[1] but be more firey when it did appear. And I requested the help of five old reeves and companions to invoke the sun (tied Jomscarf) by means of Earth (Breaburn Apple), Jupiter (Orange), Saturn (Lemon), Mars (Red pepper, plus tomato) and Moon (Mushroom, mmm), each by a variety of names of power, rather than speaking to the sun directly in one of its languages.

[1] I'd intended it to rise at 7.44, but it only cleared the horizon a couple of minutes later.

Then we drank the mead, ate the planets, and decamped to clowns, who unblinkingly provided cooked breakfast for 15, and I ran into at least one other person I knew.

Then I went home and slept and spodded for 24 hours, completely confusing my sleep cycle.

Edit: Q. So, if you don't do this, the sun doesn't come up?
A. In theory. No-one's ever been brave enough to test it.

Edit: And thanks to ceb and Mark for advice.