2019/09/25 by Panu Lahti, Lahti, Panu, Xiaodan Zhou +1
Mathematics · #26A45 #30L99 #54E35 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1909.11530
openalex publication_date 2019/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda. Furthermore, we study the necessity of conditions on the underlying space in Federer's characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincaré inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.