Diameter-minimizing non-Euclidean n-gons of fixed perimeter

Determine, in hyperbolic and spherical geometry, the convex n-gon of fixed geodesic perimeter that minimizes the geodesic diameter. Specifically, for each n ≥ 3, identify the convex hyperbolic n-gon in the Poincaré disk and the convex spherical n-gon in an open hemisphere that achieve the minimum possible diameter among all such polygons with the same perimeter.

Background

The paper proves that among hyperbolic and spherical n-gons of fixed perimeter, the regular polygon maximizes area, paralleling the classical Euclidean result. In the introduction, the authors point to a related classical isoperimetric problem already studied in Euclidean geometry: among convex Euclidean n-gons of given perimeter, minimize the diameter, as referenced in [5].

They explicitly note that this diameter-minimization problem has not been investigated in non-Euclidean (hyperbolic and spherical) geometries, thereby highlighting an unresolved question parallel to the area-maximization result established in the paper.