The isoperimetric problem

Although the isoperimetric problem is an old topic in geometry, many basic questions about it remain unsolved.  Nowadays,  Isoperimetry  is an active field of research in several areas: differential geometry, discrete and convex geometry, probability, Banach space theory, PDE...
These notes about The Isoperimetric Problem (40 pages,  pdf file, 500 kb) correspond  to a  lecture series given by the author during the

 Clay Mathematics Institute Summer School on the Global Theory of Minimal Surfaces
 June 25, 2001 to July 27, 2001, at the Mathematical Sciences Research Institute, Berkeley, California

Solving the isoperimetric problem in a region by soap bubbles experiments


1. Presentation
1.1 The isoperimetric problem for a region
1.2 Soap bubbles
1.3 Euclidean space, slabs and balls
1.4 The isoperimetric problem for the Gaussian measure
1.5 Cubes and boxes
1.6 The periodic isoperimetric problem

2. The isoperimetric problem for a Riemannian 3-manifold
2.1 The stability condition
2.2 The 3-dimensional projective space
2.3 Isoperimetry and bending energy
2.4 The isoperimetric profile
2.5 LÚvy-Gromov isoperimetric inequality

3. The isoperimetric problem for measures
3.1 The Gaussian measure
3.2 Symmetrization with respect to a model measure
3.3 Isoperimetric problem for product spaces
3.4 Sobolev-type inequalities