2019/08/02 by Otis Chodosh, Chodosh, Otis, Max Engelstein +3 · 2 citations
Engineering · Mathematics · Medicine · #Analysis of PDEs (math.AP) #Bone and Joint Diseases #Differential Geometry (math.DG) #FOS: Mathematics #Numerical methods in inverse problems #Sparse and Compressive Sensing Techniques #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.1908.00677
28 pages. Comments welcome
arxiv created 2019/08/02 · openalex publication_date 2019/08/02 · arxiv updated 2019/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) version of the quantitative isoperimetric inequality holds for a real analytic metric, using the Lojasiewicz-Simon inequality. A main novelty of our work is that in all our results we do not require any a priori knowledge on the structure/shape of the minimizers.