Area Functionals with Affinely Regular Polygons as Maximizers

1. Area Functionals with Affinely Regular Polygons as Maximizers P.Gronchi - M.Longinetti

2. P

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

4. 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)

6. 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.

7. Lagrange multipliers argument for the outer problem The maximizing polygon solves the system

8. Lagrange multipliers arguments for the outer problem

10. 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)

11. 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.

13. 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.

14. 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.

15. 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

16. 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.

17. 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.

19. Extensions to planar convex bodies Si P S C

20. Discrete Affine Length

21. 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)