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...

Solving the isoperimetric problem in a region by soap bubbles experiments
Contents
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 3manifold
2.1 The stability condition
2.2 The 3dimensional projective space
2.3 Isoperimetry and bending energy
2.4 The isoperimetric profile
2.5 LévyGromov 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 Sobolevtype inequalities