Thursday, April 07, 2011

A proof, or not a proof?

i'm relieved -- it turns out that my proof is wrong.
i should probably explain which one, though ..

last weekend at a coffeehouse, it occurred to me that a certain theorem in metrιc geοmetry [1] was exactly the right tool that i needed. i couldn't remember the proof though, and being without a computer (and mathscinet) i tried to re-prove it on my own.

by sunday, i had a good idea and thought it would be a new proof.

on monday, it occurred to me that the argument of proof is too strong: i worked through an example that would contradict(!) one of my results from this year ..
.. so i spent two days frantically checking my proof of that result:

lemma by lemma,
corollary after theorem ..
.. so far, so good!

admittedly, the proof is a little strange. (i might write about it sometime.)
yesterday i browsed through the proof of that geοmetric theοrem: the ideas are completely different from my naive ones, which is relieving ..

.. and this morning, i checked through that naive "good" idea from sunday. it doesn't work -- i see why, now -- which settles the issue:
yes, i'm an idiot;
then again, my earlier theorem is still a theorem.

as a bonus, i think i've found the right piece of evidence. that theorem (as discussed here and here) looks to be unambiguously original.

[1] it's Thm 14.2 of Cheegεr's 1999 GΛFΛ paper, regarding the rectifiabilιty of isometric images of a certain class of metric spaces. (the hypotheses are somewhat technical.)

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