Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves
Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves
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DOI：
10.4310/ajm.1999.v3.n3.a5
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发表时间：
1999
影响因子：
0.6
通讯作者：
Mark R. Green;S. Osher
中科院分区：
文献类型：
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作者：
Mark R. Green;S. Osher
0.导论.本文得到了关于曲率的凸函数积分的几个新的不等式。Wulff曲率）的凸平面曲线。我们还表明，我们的不等式的两侧之间的差异是单调递减的区域流动下的单位速度向外法线（分别。Wulff）流动。给定一个有界平面区域K，单位速度向外法向流已被广泛研究，并在燃烧等许多应用问题中引起人们的兴趣。相反，如果K通过改变向外法向速度来增长，使其成为单位法向方向的函数7（0），则具有WulfF流，这也是相当感兴趣的，例如在研究晶体的生长[O-M]中。当区域K是凸的时，有一个简单的封闭形式的表达式描述这些流动，并且该区域在第一种情况下收敛到圆盘，在第二种情况下收敛到伍尔夫形状。当初始区域K是凸的时，该区域的面积是关于t的多项式，分别称为Steiner多项式或WulfF-Steiner多项式。我们的方法的一个新的特点是，我们研究的根的施泰纳和WulffSteiner多项式，发生在负值的t。经典的等周不等式在这两种情况下指出，这些多项式有（负的）真实的根ti >t2，并且它们是不同的当且仅当K不是圆盘（相应地不是武尔夫形状）。Bonnesen不等式指出内半径和外半径r^和re位于区间[-£1，-£2]，如果K不是圆盘（分别不是武尔夫形状），则位于开区间。我们的不等式是最自然的陈述和证明的根ti和£ 2我们认为，这是一个潜在的相当富有成效的方法来研究凸体在更高的维度。在这种新方法的背景下，一个非常自然的联系，向外正常和武尔夫流和曲率积分的区域出现。在重要的情况下，我们的不等式状态是积极的数量被证明是单调递减的流动下的区域演变。特别提示的是，熵的曲率（分别武尔夫曲率）是有界以上的面积，是单调递减的时间。不平等本身就很吸引人。这是一个惊喜，我们有有趣的新的东西要说的凸平面曲线。我们陈述我们的不等式在这里任意光滑有界凸平面区域K的曲率的情况下，和离开的Wulff的情况下的身体的文件。其中之一是由于盖奇[G]，他的结果是我们的灵感来源。盖奇的结果是
0. Introduction. In this paper, we obtain some new inequalities for integrals of convex functions of the curvature (resp. Wulff curvature) of convex plane curves. We also show that the difference between the two sides of our inequalities are monotone decreasing as the region flows under the unit-speed outward normal (resp. Wulff) flow. Given a bounded plane region K, the unit-speed outward normal flow has been highly studied, and is of interest in many applied problems, e.g. combustion. If instead K grows by varying the outward normal speed to be a function 7(0) of the direction of the unit normal, one has the WulfF flow, which is also of considerable interest, e.g. in studying the growth of crystals [O-M]. When the region K is convex, there is a simple closed-form expression which describes these flows, and the region converges to a disk in the first case and a Wulff shape in the second. The area of the region when the initial region K is convex is a polynomial in t, known respectively as the Steiner polynomial or WulfF-Steiner polynomial. A novel feature of our approach is that we study the roots of the Steiner and WulffSteiner polynomials, which occur at negative values of t. The classical isoperimetric inequality in both cases states that these polynomials have (negative) real roots ti >t2, and that they are distinct if and only if K is not a disc (respectively not a Wulff shape). Bonnesen's inequality states that the inradius and outradius r^ and re lie in the interval [—£1, —£2], and in the open interval if K is not a disk (respectively not a Wulff shape). Our inequalities are most naturally stated and proved in terms of the roots ti and £2We feel that this is a potentially quite fruitful approach to studying convex bodies in higher dimensions. In the context of this new approach, a very natural link between the outward normal and Wulff flows and the curvature integrals of the region appears. In important cases, the quantities that our inequalities state are positive are shown to be monotone decreasing as the region evolves under the flow. Particularly suggestive is the fact that the entropy of the curvature (respectively Wulff curvature) is bounded above in terms of the area and is monotone decreasing with time. The inequalities themselves are quite fascinating. It came as a surprise to us that there are interesting new things to be said about convex plane curves. We state our inequalities here for arbitrary smooth bounded convex plane regions K in the curvature case, and leave the Wulff case to the body of the paper. One of them is due to Gage [G], whose result was a source of inspiration to us. Gage's result is