The principle of symmetry--extremity (Q1406080)

The author studies connections between symmetry and extreme properties. The main study in the article is on a group of automorphisms of the system \((X,S,\mu)\) where \(X\) is a separable locally compact Hausdorff measure space, \(S\) is the \(\sigma \)-ring of the Baire sets and \(\mu \) is a Baire measure defined on \(S\). The author conjectures that the higher the symmetry, the more are extreme properties. Also, a connection with the isoperimetric problem is given.