[Paper Review] Isoperimetric inequalities of euclidean type in metric spaces

Stefan Wenger|ArXiv.org|Jun 5, 2003

Geometric Analysis and Curvature Flows4 references3 citations

TL;DR

This paper establishes an isoperimetric inequality of Euclidean type in complete metric spaces that satisfy a cone-type inequality, generalizing results by Gromov and Ambrosio-Kirchheim. It proves that such spaces admit a filling of any $k$-dimensional integral current $T$ with zero boundary by a $(k+1)$-current $S$ satisfying $\mathbf{M}(S) \leq D_k \mathbf{M}(T)^{(k+1)/k}$, with $D_k$ depending only on $k$ and the space's geometric constants.

ABSTRACT

In this paper we prove an isoperimetric inequality of euclidean type for complete metric spaces admitting a cone-type inequality. These include all Banach spaces and all complete, simply-connected metric spaces of non-positive curvature in the sense of Alexandrov or, more generally, of Busemann. The main theorem generalizes results of Gromov and Ambrosio-Kirchheim.

Motivation & Objective

• To resolve the isoperimetric problem of Euclidean type in complete metric spaces that admit a cone-type inequality.
• To generalize Gromov's and Ambrosio-Kirchheim's results on isoperimetric inequalities from Riemannian manifolds and dual Banach spaces to a broader class of metric spaces.
• To answer affirmatively the open question of whether all Banach spaces admit an isoperimetric inequality of Euclidean type.
• To establish a general framework for constructing isoperimetric fillings in metric spaces using decomposition and cone-type inequalities.
• To provide explicit, computable constants $D_k$ for the isoperimetric inequality in terms of geometric and analytic parameters of the space.

Proposed method

• Utilizes the theory of metric integral currents $\mathbf{I}_k(X)$ as defined by Ambrosio and Kirchheim for surfaces in metric spaces.
• Applies a cone-type inequality: for any $k$-cycle $T$, there exists a $(k+1)$-current $S$ with $\partial S = T$ and $\mathbf{M}(S) \leq C_k \cdot \text{diam}(\operatorname{spt} T) \cdot \mathbf{M}(T)$.
• Employs an iterative decomposition of a $k$-current $T$ into smaller cycles $T_i$ with controlled diameter and mass, using a slicing technique and a parameter $\lambda$.
• Constructs a filling current $S$ as the limit of sums of cone-fillings $S_i$ of the decomposed cycles $T_i$, ensuring convergence in the mass norm.
• Uses the triangle inequality and $L^{(k+1)/k}$-summability of masses to bound the total mass of the filling $S$.
• Relies on the completeness of the space of integral currents $\mathcal{I}_{k+1}(X)$ to guarantee convergence of the Cauchy sequence $S^n = \sum_{i=1}^{N_n} S_i$ to a limit $S \in \mathbf{I}_{k+1}(X)$ with $\partial S = T$.

Experimental results

Research questions

• RQ1 Does every complete metric space admitting a cone-type inequality for $\mathbf{I}_k(X)$ also admit an isoperimetric inequality of Euclidean type for $\mathbf{I}_k(X)$?
• RQ2 Can the isoperimetric inequality of Euclidean type be established in Banach spaces, even when they are not dual or finite-dimensional?
• RQ3 What is the role of the cone-type inequality in enabling the construction of isoperimetric fillings in non-Riemannian metric spaces?
• RQ4 How does the iterative decomposition of currents into small-diameter cycles facilitate the construction of a uniformly bounded filling?
• RQ5 To what extent do spaces with $\gamma$-convex bicombings (e.g., non-positively curved spaces) satisfy the isoperimetric inequality of Euclidean type?

Key findings

• Every complete metric space admitting a $k$-dimensional cone-type inequality and an isoperimetric inequality of Euclidean type for $\mathbf{I}_{k-1}(X)$ also admits such an inequality for $\mathbf{I}_k(X)$, with a constant $D_k$ depending only on $k$, the cone constant $C_k$, and the $\mathbf{I}_{k-1}$-constant.
• All Banach spaces admit an isoperimetric inequality of Euclidean type for $\mathbf{I}_k(E)$, with universal constants $D_k$ independent of the Banach space, answering an open question of Ambrosio and Kirchheim.
• Spaces with a $\gamma$-convex bicombing (e.g., simply-connected Alexandrov or Busemann spaces of non-positive curvature) admit an isoperimetric inequality of Euclidean type with constants $D_k$ depending only on $k$ and $\gamma$.
• The isoperimetric filling $S$ of a $k$-cycle $T$ satisfies $\mathbf{M}(S) \leq C_k E \left(\frac{1+\lambda}{\delta}\right)^{(k+1)/k} \mathbf{M}(T)^{(k+1)/k}$, where $E$, $\lambda$, and $\delta$ are parameters from the decomposition process.
• The construction yields a Cauchy sequence of partial fillings $S^n = \sum_{i=1}^{N_n} S_i$ in the mass norm, converging to a limit $S \in \mathbf{I}_{k+1}(X)$ with $\partial S = T$.
• The constants $D_k$ in the main theorem and its corollaries are computable, though not optimal, even for $X = \mathbb{R}^n$.