Area Functionals with Affinely Regular Polygons as Maximizers

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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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Area Functionals with Affinely Regular Polygons as Maximizers

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Area Functionals with Affinely Regular Polygons as Maximizers

P.Gronchi - M.Longinetti

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

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)

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.

Lagrange multipliers argument for the outer problem

The maximizing polygon solves the system

Theorem 2: For any n > 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)

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.

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.

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.

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 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

are equal.

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.

Geometric Tomography

Volčič, 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 solutionsto the Hammer’s X-ray

problem in m-directions,n=2m.

A stability result:

in preparation (P.Dulio, C.Peri, A.Venturi)