The isoperimetric inequality for convex subsets of the sphere

We give a new proof of an isoperimetric inequality for family closed surfaces, which have Gaussian curvature identically equal to one wherever the surface is smooth. These surfaces are formed from convex, spherical polygon, with each vertex polygon leading non-smooth point on surface. For example, lune revolution, two tips. Combined straightforward approximation argument, this was first proved by B\'erard, Besson, and Gallot, where they provide generalization L\'evy-Gromov inequality. The implies geodesically convex subsets sphere, and, using Faber-Krahn theorem, it also lower bound Dirichlet eigenvalue region given area surfaces. Via approximation, we convert into Dirichlet-Neumann domains contained in sphere.