on a completely different note: any calc. of variations people here? i was trying to research something which i was sure i'd seen before, but haven't found anything directly like what i was looking for
Suppose I consider Dido's problem: With the perimeter held fixed, which closed plane curve encloses the largest possible area? The answer, of course, is a circle.
what i thought was that there was some way to make this precise, in the following sense: Can one construct an evolution equation for the perturbed curve (i.e. taking that as the initial condition) which will smoothly evolve to the circle?
I think Mike refers to the brand new rolling limit. It's not a uniform N per month limit, but something that depends on how previous questions were received.
I changed the denominator because the other one was wrong; if the limit is indeed 0 it doesn't matter (and asymptotics will only differ by a value of $1/32$) but this is still the one that actually counts natural density
kk. i may edit the original question to just say "hey, MikeMiller put up a question about a quantitative question about solvable v. unsolvable" rather than asking it directly, since i've already accepted an answer
@Rafflesiaarnoldii: my 'real' motivation was: given an initial plane curve (not a circle), can i define an evolution equation which preserves both area and perimeter?
the calculus of variations connection there is a lot less obvious, since there's no quantity i'm evidently extremalizing
i was more interested in finding an evolution equation which has two invariant quantities rather than just one
@TedShifrin So how many different notions of connection do I have? I already know about affine connections from Petersen's class, I'm now learning about these hip Ehresmann connections... are these the main two? The first for vector bundles and the latter for principal $G$-bundles? Am I about to learn about twenty different kinds of connection?
@Semiclassical Sorry, never saw such a thing; nor can I imagine how it could work. Preserving one parameter is done by running the standard curve-shortening flow, while simultaneously rescaling. There's a seminar talk on such flows tomorrow; I'll try to remember to ask the speaker.
the link to an evolution equation was b/c i was hoping to find a linkage to a known special function of some kind
mostly because the two-bump example, if somehow converted into something which wasn't locally isometric, seemed like it would awfully reminiscient of this:
