Abstract
We present some recent stability results concerning the isoperimetric inequality and other related geometric and functional inequalities. The main techniques and approaches to this field are discussed.
Highlights
The isoperimetric inequality is probably one of the most beautiful and at the same time one of the most powerful tools of modern mathematics
After quickly recalling some basic tools from De Giorgi’s theory of sets of finite perimeter and from Geometric Measure Theory, in Sect. 3 we present Fuglede’s proof of the quantitative isoperimetric inequality for convex sets and for nearly spherical sets, that are sets close to a ball in C1 sense
In the previous section we have presented the proof of the the quantitative isoperimetric inequality given in [76]
Summary
The isoperimetric inequality is probably one of the most beautiful and at the same time one of the most powerful tools of modern mathematics. Theorem 1.1 (Bonnesen) Given a closed, simple curve γ ⊂ R2 enclosing a convex set C of area A, there exist two concentric circles C1 ⊂ C ⊂ C2 of radii r1 and r2, respectively, such that (r2 − r1)2 This inequality has the feature of being sharp, since the constant 4π at the denominator cannot be increased, and of having an elementary proof. Volume as the unit ball B one can always measure the Hausdorff distance of a translate of K from B by a suitable power of the isoperimetric deficit of K , i.e., the difference P(K ) − P(B), see Theorem 3.2 His result was the starting point of modern investigations on the stability of isoperimetric inequality. At that point we had to stop since for some of them new developments are foreseen in the years
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