Extremal Areas of Polygons with Fixed Perimeter
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EXTREMAL AREAS OF POLYGONS WITH FIXED PERIMETER G. Khimshiashvili,∗ G. Panina,† and D. Siersma‡
UDC 515.164.174
We consider the configuration space of planar n-gons with fixed perimeter, which is diffeomorphic to the complex projective space CP n−2 . The oriented area function has the minimum number of critical points on the configuration space. We describe its critical points (these are regular stars) and compute their indices when they are Morse. Bibliography: 11 titles.
1. Introduction One of the results on the isoperimetric problem discussed in the classical treatise by Legendre [7] states that the regular n-gon has the maximum area among all n-gons with fixed perimeter. For the historic development of this problem, we refer to [2]. The aim of the present paper is to elaborate upon this classical result by placing it in the context of Morse theory on a naturally associated configuration space. To this end, we follow the paradigms used in [9, 10] and begin with several definitions and recollections. An n-gon is an n-tuple of points (p1 , . . . , pn ) ∈ (R2 )n , some of which may coincide. Its perimeter is (as usual) P(p1 , . . . , pn ) = |p1 p2 | + |p2 p3 | + · · · + |pn p1 |. The configuration space Cn considered in what follows is defined as the space of all polygons (modulo rotations and translations) whose perimeter equals 1 (one can fix any other positive number). The oriented area of a polygon with vertices pi = (xi , yi ) is defined as 2A = x1 y2 − x2 y1 + · · · + xn y1 − x1 yn . The oriented area as a Morse function has been studied in various settings: for configuration spaces of flexible polygons (those with the side lengths fixed) in R2 and R3 , see [5, 9–11]. We shall use some of the previous results in the new setting of this paper. Let σ be the cyclic renumbering: given a polygon P = (p1 , . . . , pn ), σ(p1 , . . . , pn ) = (p2 , p3 , . . . , pn , p1 ). In other words, we have an action of Zn on Cn which renumbers the vertices of a polygon. A regular star is an equilateral n-gon such that σ(p1 , . . . , pn ) = (p1 , . . . , pn ), see Figs. 1, 2. A complete fold is a regular star with pi = pi+2 . It exists for even n only. A regular star that is not a complete fold is uniquely determined by its winding number w with respect to the center. We can now formulate the main result of the paper. ∗
Ilia State University, Tbilisi, Georgia, e-mail: [email protected].
†
St.Petersburg Department of Steklov Institute of Mathematics and St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected]. ‡
Utrecht University, Utrecht, The Netherlands, e-mail: [email protected].
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 481, 2019, pp. 136–145. Original article submitted Jule 12, 2019. 1072-3374/20/2475-0731 ©2020 Springer Science+Business Media, LLC 731
Theorem 1. (1) The space Cn is homeomorphic to CP n−2 . Therefore, we regard it as a smooth manifold, keeping in mind the smooth structure of the projective space. (2) Smooth critical points of the function A on Cn are regular stars and co
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