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The isoperimetric problem on some singular surfaces. (English) Zbl 1093.49027

The authors characterize the least-perimeter enclosures of prescribed area on some convex polyhedra (cube, regular tetrahedron, regular octahedron, and rectangular prisms), on double disks of constant curvature, and on cylindrical cans. All these surfaces are singular, i.e., piecewise smooth two-dimensional manifolds. The way of proving these results is through classification of constant geodesic curvature curves. In addition, symmetrization is used in the cases of double disks and cylindrical cans to simplify the least-perimeter candidates.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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