Discrete length-volume inequalities and lower volume bounds in metric spaces
Discrete length-volume inequalities and lower volume bounds in metric spaces
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度量空间中的离散长度-体积不等式和体积下界
DOI：
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发表时间：
2014
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通讯作者：
K. Kinneberg
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作者：
K. Kinneberg
W.德里克确保任何黎曼立方体$$（[0，1]^n，g）$$（[0，1]n，g）的体积由相对余维1面之间的距离的乘积限制。本文基于度量空间中的一个体积下界问题，建立了立方[0，1]^n [0，1]n的加权开覆盖的离散Derrick不等式.我们的主要定理推广了作者在Kinneberg（J Differ Geom 100（2）：349-388，2015）中的先前结果，该结果给出了Derrick不等式的组合版本，并用于双曲群的边界分析。作为应用，我们回答了Y的一个问题. Burago和V. Zalgaller关于单位立方体上伪度量的长度-体积不等式。
A theorem of W. Derrick ensures that the volume of any Riemannian cube $$([0,1]^n,g)$$([0,1]n,g) is bounded below by the product of the distances between opposite codimension-1 faces. In this paper, we establish a discrete analog of Derrick’s inequality for weighted open covers of the cube $$[0,1]^n$$[0,1]n, which is motivated by a question about lower volume bounds in metric spaces. Our main theorem generalizes a previous result of the author in Kinneberg (J Differ Geom 100(2):349–388, 2015) which gave a combinatorial version of Derrick’s inequality and was used in the analysis of boundaries of hyperbolic groups. As an application, we answer a question of Y. Burago and V. Zalgaller about length-volume inequalities for pseudometrics on the unit cube.