Perimeter is a fundamental notion in Euclidean geometry, and the classical Euclidean isoperimetric
inequality is one of the most important results in geometry. It states that Euclidean balls have maximal
volume among sets of given perimeter. Many results in geometry and analysis rely on this result. In
particular, the so-called Sobolev inequalities, fundamental for the theory of Partial Differential Equations,
can be deduced from the Euclidean isoperimetric inequality.
At the beginning of the 19th century, Minkowski introduced anisotropic perimeters (also called first mixed
volumes) and established the Minkowski inequality (also called the anisotropic isoperimetric inequality).
This inequality has numerous applications and constitutes one of the pillars of the so-called Brunn-
Minkowski theory.
In recent years, non-local versions of perimeter have attracted a lot of attention. In this setting, the
isoperimetric inequalities state that Euclidean balls have maximal volume among sets of given s-fractional
perimeter (where s is a parameter). These and further non-local functionals are used to model non-local
phenomena in many applications.
The proposed research aims to establish non-local versions of the anisotropic isoperimetric inequality and
related inequalities. Such a result corresponds to the step from Euclidean geometry to the Brunn-
Minkowski theory in the non-local setting. The project also aims to establish corresponding inequalities in
analysis, particularly sharp anisotropic fractional Sobolev inequalities.