Area Functionals with Affinely Regular Polygons as Maximizers

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Area Functionals with Affinely Regular Polygons
as Maximizers

• P.Gronchi - M.Longinetti

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P
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Index

• Variational Arguments
• Algebraic Systems
• Extensions to planar convex bodies
• Affine Length and approximation
• Application to geometric tomography

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Proof geometric variational argument, if it is
not true there exist j
then we move a side of P and we get P such that
M.L. (1985)
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A new property for the maximizers polygons of
the outer problem
THEOREM For the maximizer polygon
P of G has the following property
where for any polygon P with sides of lenght
Note each
is an affinely invariant ratio of distances
between four aligned points similarly to the
cross-ratio.
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Lagrange multipliers argument for the outer
problem

The maximizing polygon solves the system
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Lagrange multipliers arguments for the outer
problem
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Theorem 2 For any n gt 5 the polygons P
maximizing G have both properties above, i.e.
Theorem 3
Remark 1
Remark 2
C.Fisher-R.E.Jamison ,Properties of Affinely
regular polygons,
Geometriae Dedicata 69 (1998)
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An algebraic system for the outer problem
Theorem A Let P be a maximizer of G then
and the set of the ratios
solves
the system of n equations
Remark is a circulant system of n
equations in
the unknowns
Theorem B all solutions
to the system
are trivial, i.e.
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Variational argument for the inner problem
The maximum value of F(P) is attained at P iff
P is an affinely regular polygons of n sides.
Lemma 1
Lemma 1 is equivalent to
Coxeter (1992) have proved that P is affinely
regular iff
are suitable real constants, depending only on n.
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Lemma 2 (Multipliers Lagrange argument)
At the maximizing polygon P the following
Lagrange system holds
is a system of (2n 2n) real equations in the
unknowns
which are represented from 4n real numbers.
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Lemma 3 If P solves the vector system
then the parameters
solve the following system
Lemma 4 the previous system is a circulant
system in the unknowns which can be splitted in
the following way
Proof of lemma 4 in order to get the first set
of the equations in the parameters µ we compare
Cj solved from the first group of equation
with one from the second one. To get the second
set we apply Rouché-Capelli theorems to (33)
consecutives equations of the system Sn in 5
consecutives unknowns
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Lemma 5 The only real solutions bigger than 1
of the following circulant system
are the trivial solutions
Proof of lemma 5 we define
The above circular system becomes
The goal is to prove that all
are equal.
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First proof (ten pages long) we look for the
sign of parallel differences
we get contradiction if the above difference is
non zero.
Second proof (magic ) let us consider
Suppose
then
Since four consecutive values are equal the
consecutives equations imply that all the values
are equal.
C.V.D.
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Extensions to planar convex bodies
P
Si
S
C
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Discrete Affine Length
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Geometric Tomography
Volcic, A. - Well-Posedness of the Gardner -Mc
Mullen Reconstructrion Problem
Oberwolfach 1983
M.L. An isoperimetric inequality for convex
polygons and convex sets with the same
symmetrals, Geometriae Dedicata 20(1986)
is related to the Nykodim distance between any
two possible solutions to the
Hammers X-ray problem in m-directions, n2m.
A stability result
in preparation (P.Dulio, C.Peri, A.Venturi)