Many figures of merit have the same regular N-gon as the optimum.
A common one when talking about the isoperimetric inequality,
for any rectifiable simple closed curve C in the plane, is
f_2(C) := length(C)^2 / area(C)
and this figure of merit f_2(C) is dimensionless, so independent of units. Then
the inequality is just
f_2(C) <= 4
with equality only for the circle.
***>>> I think this is a fascinating area to explore. It may be that nothing
too interesting happens in 2D. But in 3D, where a corresponding figure of
merit for a rectifiable closed surface S in R^3 could be
f_3(S) = area(S)^3 / volume(S)^2,
and then the inequality is
f_3(S) <= 36
again with equality only for the 2-sphere.
Then for any given combinatorial type of polyhedron (the equivalence class of
polyhedra having 1-1 correspondences between their faces of each dimension
preserving incidence), we can ask for the one(s) that maximize f_3(S).
A figure-of-merit with similar properties is the ratio of the radius of the
inscribed sphere to that of the circumscribed sphere:
R_inscr / R_circum <= 1
(again with equality only for the sphere; this also holds in n dimensions).
Another figure-of-merit might be to try to make the vertices as far apart as
possible, again among polyhedra of the same type. In 3D one way is to sum up
all squared distances between points, with distances reckoned through 3-space.
This kind of thing would not work if there existed no stationary point (for
the figure of merit) within the given combinatorial type. But making the vertices
as far apart as possible tends to make them equal, so this is OK.
—Dan
Allan Wechsler wrote:
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For a fixed N, suppose we want to make a polygonal pasture with N sides,
with the largest possible area given a fixed perimeter. Is the unique
optimum always a regular N-gon? I am looking for a figure of merit that
tells how regular a polygon is, and perimeter/area seems promising.
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