A note on Hurwitz's inequality
A note on Hurwitz's inequality
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DOI：
10.1016/j.jmaa.2017.09.017
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发表时间：
2017-04
影响因子：
1.3
通讯作者：
J. Cuf'i;E. Gallego;A. Revent'os
中科院分区：
文献类型：
--
作者：
J. Cuf'i;E. Gallego;A. Revent'os
给定一个长度为L的简单闭平面曲线Γ，它包含一个面积为F的紧凸集K，Hurwitz发现了等周亏格的上界，即L2 − 4 π F≤ π| F e|其中Fe是由Γ的渐屈线包围的代数面积。本文改进了这个不等式，找到了亏额π的严格正下界|F e| − Δ，其中Δ= L 2− 4 π F。这些界限涉及到r的视角或K相对于K的Steiner点的踏板曲线或K与K的Steiner圆盘之间的L2距离。对于常宽紧凸集，Hurwitz不等式可改进为L2 − 4 π F≤ 4 9 π| F e|.在这种情况下，我们也得到了亏损4 9 π的严格正下界|F e|- Δ。对于每个已建立的不等式，我们研究等式何时成立。这发生在那些紧凸集是由一条曲线平行于内摆线的3，4或5尖或闵可夫斯基和这类集。
Given a simple closed plane curve Γ of length L enclosing a compact convex set K of area F, Hurwitz found an upper bound for the isoperimetric deficit, namely L 2− 4 π F≤ π| F e|, where F e is the algebraic area enclosed by the evolute of Γ. In this note we improve this inequality finding strictly positive lower bounds for the deficit π| F e|− Δ, where Δ= L 2− 4 π F. These bounds involve either the visual angle of Γ or the pedal curve associated to K with respect to the Steiner point of K or the L 2 distance between K and the Steiner disk of K. For compact convex sets of constant width Hurwitz's inequality can be improved to L 2− 4 π F≤ 4 9 π| F e|. In this case we also get strictly positive lower bounds for the deficit 4 9 π| F e|− Δ. For each established inequality we study when equality holds. This occurs for those compact convex sets being bounded by a curve parallel to an hypocycloid of 3, 4 or 5 cusps or the Minkowski sum of this kind of sets.