Quantitative isoperimetric inequalities in the C-GNP setting

Develop quantitative versions of the Faber–Krahn, Szegő–Weinberger, and Payne–Rayner type isoperimetric inequalities established for level sets and domains in O_C, providing stability or deficit estimates.

Background

The paper proves several isoperimetric-type inequalities in the C-GNP framework using coarea-based arguments and Steiner symmetrization adapted to C. These results are currently qualitative in nature.

The authors explicitly ask whether quantitative refinements—providing stability or deficit bounds relative to optimal shapes—can be established in this geometric class.