2020/12/04 by Thierry De Pauw, De Pauw, Thierry, Robert Hardt +1 · 1 citation
Mathematics · #14P10 #32B20 #49J45 #49Q15 #49Q20 #52A40 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2012.02667
openalex publication_date 2020/12/04 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
The isoperimetric inequality for a smooth compact Riemannian manifold A provides a positive \bf c(A), so that for any k+1 dimensional integral current S0 in A there exists an integral current S in A with ∂ S=∂ S0 and \bf M(S)≤ \bf c(A)\bf M(∂ S)(k+1)/k. Although such an inequality still holds for any compact Lipschitz neighborhood retract A, it may fail in case A contains a single polynomial singularity. Here, replacing (k+1)/k by 1, we find that a linear inequality \bf M(S)≤ \bf c(A)\bf M(∂ S) is valid for any compact algebraic, semi-algebraic, or even subanalytic set A. In such a set, this linear inequality holds not only for integral currents, which have \boldsymbolZ coefficients, but also for normal currents having \boldsymbolR coefficients and generally for normal flat chains with coefficients in any complete normed abelian group. A relative version for a subanalytic pair B⊂ A is also true, and there are applications to variational and metric properties of subanalytic sets.