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Topic: Fasbender's Theorem
Replies: 4   Last Post: Nov 7, 1995 8:56 PM

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Dr. M.A.B. Deakin

Posts: 15
Registered: 12/3/04
Fasbender's Theorem
Posted: Oct 15, 1995 7:34 PM
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I recently had a query on a result known (correctly) as "Fasbender's
Theorem". It came as a result of its being set as a topic of special study
in our state high schools here, but none of the examiners could enlighten
me as to its provenance. The best I could get (from one of our most senior
maths teachers here) were two vague and (as it happened) incorrect
references. (One might meditate on the state of history of maths in our
schools!) However, by diligent search I found out a lot and am hoping that
someone "out there" can tell me more.

The theorem concerns what I shall call "floppy n-gons"; these have their
sides prescribed, but not their angles. We now have:

*A floppy n-gon has maximal area when its vertices are concyclic.*

The theorem was first stated by Hermann Umpfenbach in Crelle 25 (1843), pp.
184-185, together with a proof of the case n=5. (Incidentally, he uses, as
standard, many formulae now quite lost, and whose proof took me some time.)

The case n=4 must have been known from antiquity. For a simple proof see
Chapter 2 of Durell & Robson's Advanced Trigonometry (London: Bell, 1930).
Also in 1843, Eduard Fasbender gave a less expeditious proof (Crelle 26,
pp. 181-182) of this case but then went on to use it to prove the general
case as a ready corollary.

Fasbender's proof of the general result is now widely available as the
final paragraph of a treatment of the problem by Ivan Niven (Maxima and
Minima without Calculus; MAA, 1981, pp. 236-237).

3 queries to complete the story:

1. As Niven does not name Fasbender, where are there references that do
give the attribution preserved correctly in the folk-lore?

2. Does anyone have a *simple & elementary* proof that a general floppy
n-gon *can always* be made to lie with its vertices on a circle?

3. In modern terms, Fasbender's theorem tells us that a finite element
approach to the classical isoperimetric problem produces an optimal result.
Is this known in the literature of the finite element method?

Thanks for any contributions. Mike Deakin

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