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The isoperimetric problem. (English) Zbl 1125.49034
Hoffman, David (ed.), Global theory of minimal surfaces. Proceedings of the Clay Mathematics Institute 2001 summer school, Berkeley, CA, USA, June 25–July 27, 2001. Providence, RI: American Mathematical Society (AMS). Cambridge, MA: Clay Mathematics Institute (ISBN 0-8218-3587-4/pbk). Clay Mathematics Proceedings 2, 175-209 (2005).
Summary: The isoperimetric problem is an active field of research in several areas, such as in differential geometry, discrete and convex geometry, probability, Banach spaces theory and PDEs. In this section we will consider some situations where the problem has been completely solved and some others where it remains open.
For the entire collection see [Zbl 1078.53002].

49Q20 Variational problems in a geometric measure-theoretic setting
53C65 Integral geometry
49Q15 Geometric measure and integration theory, integral and normal currents in optimization