Quantitative Isoperimetric Inequalities in Geometric Measure Theory
Summary
Quantitative isoperimetric inequalities strengthen the classical statement that among all sets of given volume, the ball minimises surface area by providing explicit estimates on how close a nearly optimal set must be to a ball. In geometric measure theory, these results rely on fine analysis of sets of finite perimeter and their deviation from spherical symmetry. The central idea is to bound a perimeter deficit—the excess boundary measure over that of a ball—by a power of a suitable asymmetry metric. This quantitative stability yields not only uniqueness and rigidity results but also rates of convergence in variational problems involving curvature flows, capillarity phenomena and phase transitions. Recent methodological advances have employed optimal transport, mass rearrangement and concentration–compactness techniques to treat anisotropic weights, curved ambient spaces and singular perturbations. Applications span from crystal shape in materials science to Sobolev embeddings in functional inequalities, underscoring the global significance of quantitative isoperimetry as a tool for understanding stability and rigidity across analysis, geometry and applied mathematics.
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Quantitative Isoperimetric Inequalities in Geometric Measure Theory publication trend
The graph below shows the total number of articles in quantitative isoperimetric inequalities in geometric measure theory across all publications each year (not limited to Nature Index journals).
Technical terms
Quantitative isoperimetric inequality: an estimate relating the perimeter excess of a set to a power of its deviation from a minimiser.
Perimeter deficit: the difference between the perimeter of a given set and the minimal perimeter for a set of the same volume.
Fraenkel asymmetry: a measure of how far a set is from a ball of equal volume, defined via the symmetric difference.
Isoperimetric profile: a function giving the minimal perimeter required to enclose a given volume in a specified ambient space.
Convex body: a compact convex subset of Euclidean space with non-empty interior.
References
- On the Three-Dimensional Shape of a Crystal. Mathematics (2025).
- The quantitative isoperimetric inequality and related topics. Bulletin of Mathematical Sciences (2015).
- Total positive curvature and the equality case in the relative isoperimetric inequality outside convex domains. Calculus of Variations and Partial Differential Equations (2023).
- Some Isoperimetric Inequalities in the Plane with Radial Power Weights. The Journal of Geometric Analysis (2023).
- Rigidity and large volume residues in exterior isoperimetry for convex sets. Advances in Mathematics (2024).
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