Browsing by Author "Arti Sahu Gangopadhyay"

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The isoperimetric problem in Randers plane
(Institute of Mathematics, University of Debrecen, 2024) Arti Sahu Gangopadhyay; Ranadip Gangopadhyay; Hemangi Madhusudan Shah; Bankteshwar Tiwari
In 1947, Busemann observed that a Minkowski circle need not be a solution of the isoperimetric problem in a Minkowski plane. Li and Mo recently showed that the Euclidean circles centred at the origin in a unit ball with the Funk metric are solutions of the isoperimetric problem [9]. In this paper, we construct a class of Randers planes in which any Euclidean circle, centered at the origin in R2, turns out to be a local minimum of the isoperimetric problem with respect to the various well-known volume forms in Finsler geometry. As a consequence, it turns out that the Euclidean circles centred at the origin are solutions of the isoperimetric problem in a Randers type Minkowski plane. © 2024 Institute of Mathematics, University of Debrecen. All rights reserved.
The isoperimetric problem in Randers Poincaré disc
(World Scientific, 2024) Arti Sahu Gangopadhyay; Ranadip Gangopadhyay; Hemangi Madhusudan Shah; Bankteshwar Tiwari
It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this paper, we show that the circles centered at origin in the Randers Poincaré disc satisfy the isoperimetric equality with respect to different volume forms however, these Randers metrics do not necessarily have constant (negative) flag curvature. In particular, we show that Osserman's result [12] of the Riemannian case cannot be extended to the Finsler geometry as such. © 2024 World Scientific Publishing Company.
The isoperimetric problem in Randers Poincaré disc
(World Scientific, 2025) Arti Sahu Gangopadhyay; Ranadip Gangopadhyay; Hemangi Madhusudan Shah; Bankteshwar Tiwari
It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this paper, we show that the circles centered at origin in the Randers Poincaré disc satisfy the isoperimetric equality with respect to different volume forms however, these Randers metrics do not necessarily have constant (negative) flag curvature. In particular, we show that Osserman's result [12] of the Riemannian case cannot be extended to the Finsler geometry as such. © World Scientific Publishing Company.