Science > Physics > Independent/Dependent Phases 14: Generalized Isoperimetric Inequalities
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Science > Physics |
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"OsherD" |
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11 Oct 2005 12:41:35 AM |
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Independent/Dependent Phases 14: Generalized Isoperimetric Inequalities |
From Osher Doctorow
An entire branch of research in mathematics and its applications is
related to Levi concentration measures of part 13 of this thread,
namely generalized isoperimetric inequalities.
One of the clearest sites is
http://www.cut-the-knot.org/do_you_know/isoperimetric.shtml,
"Isoperimetric theorem and inequality," listed as copyright 1996-2005
Alexander Bogomolny who in fact references one of his published papers
on the topic ("On the perimeter and area of fuzzy sets," international
Journal of Fuzzy Sets and Systems 23, 1987, 257-269.
In 2 and 3 dimensional Euclidean spaces, Bogomolny gives the
(generalized) isoperimetric inequalities as:
1) 4piA < = L^2 (A area, L perimeter, of planar object)
2) 36piV^2 < = S^3 (S surface area, V volume of 3-d body)
The second is equivalent to: for all 3 dimensional solids with a given
surface area, the sphere has the largest volume, while the first says
that for all 2 dimensional objects with the same perimeter the largest
area is that of the circle (although when the areas are the same, the
circle has shortest perimeter).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 14: Generalized Isoperimetric Inequalities |
11 Oct 2005 01:23:59 AM |
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From Osher Doctorow
If we write:
1) 36piV^2/V_s^2 < = S^3/S_S^3
where V and S are the volume and surface area of an arbitary
3-dimensional object and V_s, S_s are the same respectively for a
sphere, and if we insert (4/3)pir^3 for V_s and 4pir^2 for S_s, we get:
2) V^2/S^3 < = 1/(36pi)
which is precisely:
3) 36piV^2 < = S^3
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 14: Generalized Isoperimetric Inequalities |
11 Oct 2005 01:40:44 AM |
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From Osher Doctorow
Take a look at Anton Petrunin's "Harmonic functions on Alexandrov
spaces and their applications," Electronic Research Announcements of
the American Mathematical Society, Volume 9, pages 135-141, Dec. 17,
2003, for a generalization of Levi's or Gromov-Levi's isoperimetric
inequality on the sphere.
Also see Sasha Sodin "Tail-sensitive Gaussian asymptotics for marginals
of concentrated measures in high dimensions," math.MG/0501382 v2 12 Aug
2005 for Levy's isoperimetric inequality on the sphere related to other
things.
Almut Burchard and Michael Schmuckenschlager (respectively U. Virginia
Math. Dept. and Institut fur Analysis und Numerik Johanes Kepler
Universitat Linz Austria), "Comparison theorems for exit times," Dec.
27, 2000, http://www.math.virginia.edu/~ab4v/preprints (40 pages).
Wikipedia and Wolfram have some related material.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 14: Generalized Isoperimetric Inequalities |
11 Oct 2005 02:10:36 AM |
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From Osher Doctorow
All this seems like a long way from f(x, y) > 1/2 or fX(x) > 1/2, but
the key word here is "sphere" or "ball".
Remember that in the last few threads I've been arguing that the
Universe changed from negative curvature to spherical curvature in its
acceleration(s).
So we can arguably say that the Universe is seeking to maximize its
volume (sphere) or minimize its surface area (sphere) with "all else
constant". A few of the papers that I've cited do in fact relate these
topics to least action.
All this puts a new "light" on tendencies to form planetary and stellar
and galactic shapes, tendencies to form life shapes, tendencies to form
virus shapes, tendencies to form condensates, etc. Admittedly it is
rather difficult to figure out what a "hurricane is trying to
maximize", or an "earthquake is trying to maximize," to put a rather
anthropomorphic phrasing on these, but on the other hand maybe "solving
them" is a matter of somehow making them spherically shaped more or
less. I've heard of worse ideas. And maybe the opposite is true for
viruses - make them non-spherical (recall the spherical minelike shape
with prongs of the HIV virus). Or make them more spherical by getting
rid of their prongs (well, who knows?). Come to think of it, the
prongs are critical to their latching onto other objects if memory
serves me right.
This reminds me of The Wizard of Oz, where concern was mainly with
where tornadoes go rather than with where they come from - but come to
think of it, where do they actually come from? Maybe we should
maintain a 24-hour watch on volcanoes, sea storms of lesser magnitude,
earthquakes or earthquake-prone regions, etc. Incidentally, the author
of Wizard of Oz, Baum, was a very smart person and is studied in
psychology and social sciences.
Then of course there's Hitchhiker's Guide to the Galaxy with its
hippo-like but enormous-nosed Judges. Well, there you have it. With
its galactic wit and irreverence toward bureaucracy and
open-mindedness, it definitely should win a Spherical Oscar :>) And
maybe it has. Sometime. Somewhere.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 14: Generalized Isoperimetric Inequalities |
11 Oct 2005 02:33:40 AM |
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From Osher Doctorow
I hesitate to mention this, but rather than have a few more millenia go
by without anybody noticing it consciously, let me ask:
Question. What's the opposite of a spherical object intuitively?
Answer. A pointed object (cone) (intuitively).
Examples of pointed objects: prongs on HIV viruses, tornadoes, volcanic
cones more or less, at least one male sexual organ more or less, at
least one female "sex-related organ" more or less, pencils and pens,
pyramids (but not the sphinx!), spears, arrows, knives, black holes
more or less, "bureaucratic pyramids". So do hurricanes start as
pointed objects? And is violence-orientation related to pointed
objects? By the way, the internet is not a pointed object. Thank God
for small favors. Oops! I just remembered, the U.N. and E. U. are
trying to take over the internet. Don't tell me that they realize how
endangered they are by the internet :>)
Osher Doctorow
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