BONNESEN-STYLE ISOPERIMETRIC INEQUALITIES
BONNESEN-STYLE ISOPERIMETRIC INEQUALITIES
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DOI：
10.1080/00029890.1979.11994723
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发表时间：
1979
影响因子：
0.5
通讯作者：
R. Osserman
中科院分区：
文献类型：
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作者：
R. Osserman
由于性质1，任何Bonnesen不等式都蕴含等周不等式（1）。根据性质2，只有当C是一个圆时，等式（1）才成立。性质3的作用是给出曲线“偏离圆度”的度量。“我们在这里的目的是，首先，回顾一下已知的平面域。特别地，我们包括形式（2）的十个不同的不等式，它们都具有Bonnesen不等式的性质1，因此包含等周不等式。这些不等式已经由不同的作者使用各种方法获得。我们表明，事实上九个十遵循一个基本的方式从一个基本的不平等，只有最后需要一个单独的证明。接下来，我们注意到，某些不等式更普遍地适用于曲面上的区域，只要高斯曲率在任何地方都是正的。最后，我们证明了这些不等式推广到任意曲面。为了详细说明最后一点，让我们注意曲面上区域的等周问题的一种形式。设D是面积为A的单连通区域，其边界为一条长度为L的简单闭曲线。假设高斯曲率K在D上有一个常数M。设D'是常曲率M的完备单连通曲面S上的测地圆盘，选择半径使D'的面积等于D的面积A。(Thus如果M > 0，则S是球面;如果M=0，则S是平面;如果M L '，则S是（直到常数因子）双曲平面，等式当且仅当D与D'等距。由于常曲率曲面上的测地圆的长度和所围的面积都很容易计算，因此不等式L > L'可以显式表示。它的形式是
Because of Property 1, any Bonnesen inequality implies the isoperimetric inequality (1). From Property 2, it follows that equality can hold in (1) only when C is a circle. The effect of Property 3 is to give a measure of the curve's "deviation from circularity." Our purpose here is, first, to review what is known for plane domains. In particular, we include ten different inequalities of the form (2), all of which have Property 1 of Bonnesen's inequality and hence imply the isoperimetric inequality. These inequalities have been obtained by various authors using a variety of methods. We show that in fact nine of the ten follow in an elementary fashion from one basic inequality; only the last needs a separate proof. Next we note that certain of the inequalities given hold more generally for domains on curved surfaces, provided the Gauss curvature is nowhere positive. Finally, we show that certain of these inequalities generalize to arbitrary curved surfaces. To elaborate on this last point, let us note one form that the isoperimetric problem can take for domains on surfaces. Let D be a simply connected domain of area A, bounded by a simple closed curve of length L. Suppose that the Gauss curvature K is bounded above on D by a constant M. Let D' be a geodesic disk on the complete simply connected surface S of constant curvature M, and choose the radius so that the area of D' equals the area A of D. (Thus S is a sphere if M > 0, S is the plane if M=0, and S is (up to a constant factor) the hyperbolic plane if M L', with equality if and only if D is isometric to D'. Since the length of a geodesic circle on a constant curvature surface and the area enclosed are both easily calculated, the inequality L > L' can be written explicitly. It takes the form