lipman@decwrl.UUCP (02/16/84)
Message-Id: <8402162035.AA28987@decwrl.uucp>
Date: Thursday, 16 Feb 1984 12:39:40-PST
From: hare::stan (Stanley Rabinowitz)
To: net.math
Subject: Maximizing A/P for curve in square
In reference to the problem that asked for the curve in the unit square
that maximized the area to perimeter ratio:
Let r be (2 sqrt(pi) - 4)/(pi-4).
The desired curve starts at (1,0) and proceeds straight up a distance 1-r.
From there is proceeds along the arc of a quarter circle to (1-r,1).
The center of the circle is at (1-r,1-r) and the radius is r.
From there it proceeds directly left to (0,1).
The maximal A/P is (2 sqrt(pi)-4)/(pi-4), which is the same as r,
and is numerically about 0.5301589042686188.
A proof that this is the optimal curve can be found in [1].
In our problem, it is clear that the curve has to be convex.
Reflecting in the axes shows that the given problem is equivalent
to finding the simple closed convex curve that lies within
a 2 X 2 square and maximizes A/P.
In fact, Lin [1] has a more general result:
Let G be a simple closed curve of perimeter P and area A contained
in a polygon of perimeter p which is circumscribable on a circle of
circumference c. Then the maximum value of A/P is achieved when
P=sqrt(cp), and this maximal value is c sqrt(p)/(2 pi (sqrt(p)+sqrt(c) ).
In our problem, the symmetrized problem is the case where the polygon
is a square with perimeter 8.
A (more difficult) proof for the square case can be found in [2].
The analogous problem when the figure is a rectangle is an unsolved
problem (see [3]).
References
[1] Tung-Po Lin, Maximum Area Under Constraint. Mathematics Magazine.
50(1977)32-34.
[2] L. A. Graham, Ingenious Mathematical Problems and Methods. Dover,
New York: 1959, problem 47, pp 29, 169-173.
[3] Singmaster and Soupporis, A Constrained Isoperimetric Problem.
Proceedings of the Cambridge Philosophical Society. 83(1978)73-82.
[4] R. F. DeMar, A Simple Approach to Isoperimetric Problems in the Plane.
Mathematics Magazine 48(1975)1-12.
[5] Bob Osserman, The Isoperimetric Inequality. Bulletin of the AMS.
84(1978)1132-1238.
Stanley Rabinowitz
...{decvax,ucbvax,allegra}!decwrl!rhea!hare!stan