Affinely Regular Polygons as Extremals of Area Functionals |
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| Authors: | Paolo Gronchi Marco Longinetti |
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| Institution: | (1) Dipartimento di Matematica e Applicazioni per l’Architettura, Università degli Studi di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy;(2) Dipartimento di Ingegneria Agraria e Forestale, Università degli Studi di Firenze, P. le delle Cascine 15, 50139 Firenze, Italy |
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| Abstract: | For any convex n-gon P we consider the polygons obtained by dropping a vertex or an edge of P. The area distance of P to such (n−1)-gons, divided by the area of P, is an affinely invariant functional on n-gons whose maximizers coincide with the affinely regular polygons. We provide a complete proof of this result.
We extend these area functionals to planar convex bodies and we present connections with the affine isoperimetric inequality
and parallel X-ray tomography. |
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| Keywords: | Affinely regular polygons Geometric tomography Affine length |
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