For regular n-gons with fixed total area, a single polygon minimizes total perimeter in Euclidean and spherical geometry, and in hyperbolic geometry only when the interior angle is at least a threshold Θ(n).
Bezdek, Ein elementarer Beweis f¨ ur die isoperimetrische Ungleichung in der Euklidischen und hy- perbolischen Ebene, Ann
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Isoperimetric Inequality for Disconnected Regions
For regular n-gons with fixed total area, a single polygon minimizes total perimeter in Euclidean and spherical geometry, and in hyperbolic geometry only when the interior angle is at least a threshold Θ(n).