A quantitative version of the isoperimetric inequality: the anisotropic case

where E ranges among all sets of finite perimeter satisfying the constraint LN (E) = const., νE is the generalized inner normal to E and ∂∗E is the reduced boundary of E (see the definitions in Section 2). For an anisotropic function Γ, one of the first attempts to solve this problem is contained in a paper by G.Wulff [22] dating back to 1901. However, it was only in 1944 that A.Dinghas [9] proved that within the special class of convex polytopes the minimiser of (1.1) is a set homothetic to