In this paper we introduce the concepts of hyperbolic and elliptic areas and prove uncountably many new geometric isoperimetric inequalities on the surfaces of constant curvature.

Keywords:

Gaussian curvature, Gauss-Bonnet theorem, polygon, pseudo-polygon, pseudo-perimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalities Gaussian curvature, Gauss-Bonnet theorem, polygon, pseudo-polygon, pseudo-perimeter, hyperbolic surface, Heron's formula, analytic and geometric isoperimetric inequalities

MSC Classifications:

51M10, 51M25, 52A40, 53C20 show english descriptions Hyperbolic and elliptic geometries (general) and generalizations
Length, area and volume [See also 26B15]
Inequalities and extremum problems
Global Riemannian geometry, including pinching [See also 31C12, 58B20] 51M10 - Hyperbolic and elliptic geometries (general) and generalizations
51M25 - Length, area and volume [See also 26B15]
52A40 - Inequalities and extremum problems
53C20 - Global Riemannian geometry, including pinching [See also 31C12, 58B20]

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