Another way to prove the triangle inequality for quarter-girths, i.e. L_{xz} <= L_{xy} + L_{yz}, is to observe that orthogonal projection onto the xz-plane does not affect L_{xz}, and cannot increase L_{xy} or L_{yz}. Consequently, it suffices to prove that the sum of the semi-axes of an ellipse is >= the quarter-circumference. That's an immediate corollary of the 'boundaries of nested convex bodies' lemma applied to the ellipse and its bounding box. -- APG.
Sent: Wednesday, June 26, 2019 at 3:43 PM From: "Brad Klee" <bradklee@gmail.com> To: "Warren D Smith" <warren.wds@gmail.com>, math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] ellipsoid from girths: multivariate series reversion
Hi Warren,
I'm busy this morning, so don't have much more time to over-explain.
Should I also have to say that projection is through the ellipse centroid, that the length variables L1', L2', L3' are also quarter-girths, and that the "flattening" procedure applies only to the 1-boundary, not also to the interior surface area? The meaning should be apparent in context, but without figures maybe not.
I would think they do these sort of constructions all the time at the better schools on the east coast (MIT, Princeton, Cornell etc...). Is it rather that all the time and money is for them to teach you how to insult lesser language so that you can then reject perfectly acceptable and insightful work?
I looked over the post again, and yes I do think it a "proof" of an "obvious" theorem. However, I am well aware that a real proof depends on an agreement between mathematicians, so hope that you or someone else with the credentials will make a genuine effort to try and figure it out, instead of just nit-picking word choice.
If you find any serious faults, please let me know.
--Brad
On Wed, Jun 26, 2019 at 9:04 AM Warren D Smith <warren.wds@gmail.com> wrote:
I don't understand your "proof."
E.g what does "can be flattened into plane" mean? What does "curvilinear projection" mean?
E.g. at least the way I normally use the word, a spherical triangle cannot be "flattened."
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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