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Area Functionals with Affinely Regular Polygons as Maximizers. P.Gronchi - M.Longinetti. P. Index. Variational Arguments Algebraic Systems Extensions to planar convex bodies Affine Length and approximation Application to geometric tomography.
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P.Gronchi - M.Longinetti
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
THEOREM: For the maximizer polygon P* of G has the
where for any polygon P with sides of lenght
is an affinely invariant ratio of distances between four aligned points similarly to the cross-ratio.
The maximizing polygon solves the system
Theorem 2: For any n > 5 the polygons P* maximizing G have both properties above, i.e. :
Remark 2 :
C.Fisher-R.E.Jamison ,Properties of Affinely regular polygons,
Geometriae Dedicata 69 (1998)
Theorem A: Let P be a maximizer of G then
and the set of the ratios
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.
The maximum value of F(P) is attained at P* iff P*is an affinely regular polygons of n sides.
Lemma 1 is equivalent to
Coxeter (1992) have proved that P* is affinely regular iff :
are suitable real constants, depending only on n.
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.
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 solvedfrom the first group of equation with one from the second one. To get the second set we apply Rouché-Capelli theorems to (3+3) consecutives equations of the system Sn in 5 consecutives unknowns
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
we look for the sign of parallel differences
we get contradiction if the above difference is non zero.
Second proof (magic ): let us consider
Since four consecutive values are equal the consecutives equations imply that
all the values are equal.
Volčič, A. - Well-Posedness of the Gardner -Mc Mullen Reconstructrion Problem
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 solutionsto the Hammer’s X-ray
problem in m-directions,n=2m.
A stability result:
in preparation (P.Dulio, C.Peri, A.Venturi)