西南大学学报 (自然科学版)  2019, Vol. 41 Issue (2): 70-74.  DOI: 10.13718/j.cnki.xdzk.2019.02.011
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  • 平面上的逆Bonnesen型Minkowski不等式    [PDF全文]
    周媛1, 张增乐2     
    1. 西南大学 数学与统计学院, 重庆 400715;
    2. 重庆文理学院 数学与财经学院, 重庆 永川 402160
    摘要:研究了平面中形如AK, L2-AKALUK, L的Minkowski不等式的上界,即逆Bonnesen-型Minkowski不等式.设K, L是平面中的凸体,其面积分别为AKAL,其中AK, L为两凸体的混合面积,UK, L为与K, L有关的几何不变量.利用平面上给定两凸体的支持函数,构造一类与给定凸体相关的新凸体.通过对新凸体几何性质的讨论,得到了一些新的加强的逆Bonnesen-型Minkowski不等式,并用此类不等式可推出一些已有结果.
    关键词凸体    支持函数    Minkowski不等式    逆Bonnesen-型Minkowski不等式    

    对于欧氏平面${{\mathbb{R}}^{2}}$中的点集K,如果对任意xyK,0≤λ≤1,都有λx+(1-λ)yK,则称K为凸集.具有非空内点的紧凸集称为凸体.若凸体K的边界∂K的曲率半径ρ>0,则称凸体K为卵形域.如果K${{\mathbb{R}}^{2}}$中的有界凸集,则以2π为周期的周期函数p(θ)称为K的支持函数. p(θ)是某个卵形域K的支持函数的充要条件是

    $ \begin{array}{*{20}{c}} {\rho = p\left( \theta \right) + p''\left( \theta \right) > 0}&{0 \le \theta < 2{\rm{ \mathsf{ π} }}} \end{array} $

    凸体K的周长、面积积分公式分别为:

    $ \begin{array}{*{20}{c}} {P = \int_0^{2{\rm{ \mathsf{ π} }}} {p{\rm{d}}\theta } }&{A = \frac{1}{2}\int_{\partial K} {p{\rm{d}}s} = \frac{1}{2}\int_0^{2{\rm{ \mathsf{ π} }}} {p\left( {p + p''} \right){\rm{d}}\theta } } \end{array} $

    凸体K, L的面积分别记为AKAL,支持函数分别为pKpL.在Minkowski加法下,凸体K+L的面积表达式为

    $ {A_{K + L}} = \frac{1}{2}\int_0^{2{\rm{ \mathsf{ π} }}} {\left( {{p_K} + {p_L}} \right)\left( {{p_K} + {p_L} + {{p''}_K} + {{p''}_L}} \right){\rm{d}}\theta } = {A_K} + {A_L} + 2{A_{K,L}} $

    其中AK, L表示两凸体的混合面积,其对应表达式为

    $ {A_{K + L}} = \frac{1}{2}\int_0^{2{\rm{ \mathsf{ π} }}} {{p_K}\left( {{p_L} + {{p''}_L}} \right){\rm{d}}\theta } = \frac{1}{2}\int_0^{2{\rm{ \mathsf{ π} }}} {{p_L}\left( {{p_K} + {{p''}_K}} \right){\rm{d}}\theta } $

    等周不等式是著名的几何不等式之一,它是最早用基本几何不变量来刻画平面几何图形的几何不等式.经典等周不等式的几何意义是:平面上周长固定的简单闭曲线中,圆所围成的面积最大.目前,等周不等式已推广到高维欧氏空间、常曲率曲面中,并应用到其他的数学分支(参见文献[1-12]).其中Minkowski不等式是经典等周不等式的推广之一.

    本文主要研究平面上的逆Bonnesen-型Minkowski不等式,形如

    $ A_{K,L}^2 - {A_K}{A_L} \le {U_{K,L}} $ (1)

    其中UK, L是与K, L有关的非负几何不变量.设K, L为欧氏平面${{\mathbb{R}}^{2}}$中的卵形域,其支持函数分别为pKpL.通过利用KL的支持函数,构造了一类新凸体Mt,其支持函数为:

    $ \begin{array}{*{20}{c}} {{p_{{M_t}}} = \left( {1 - t} \right){p_K} + tC{p_L}}&{0 \le t \le 1} \end{array} $ (2)

    其中C为与K, L相关的几何不变量,tC取值的不同而改变其取值范围的参数.当t=0时,Mt为凸体K;当C=1,t=1时,Mt为凸体L.设:

    $ \left\{ \begin{array}{l} {\vartheta _m}\left( {K,L} \right) = \min \left\{ {\frac{{{\rho _K}\left( \theta \right)}}{{{\rho _L}\left( \theta \right)}}:0 \le \theta \le 2{\rm{ \mathsf{ π} }}} \right\}\\ {\vartheta _M}\left( {K,L} \right) = \max \left\{ {\frac{{{\rho _K}\left( \theta \right)}}{{{\rho _L}\left( \theta \right)}}:0 \le \theta \le 2{\rm{ \mathsf{ π} }}} \right\} \end{array} \right. $ (3)

    $C=-{{\vartheta }_{m}}\left(K, L \right), ~0 < t\le \frac{1}{2}$时,和当$C=-{{\vartheta }_{M}}\left(K, L \right), \text{ }\frac{1}{2}\le t < 1$时,我们确定新凸体Mt,通过讨论凸体Mt的几何性质,我们得到一些新的逆Bonnesen-型Minkowski不等式,并且加强了文献[12]中定理4.6的结果,同时也给出了等号成立条件.

    引理1  设K, L为欧氏平面${{\mathbb{R}}^{2}}$中的卵形域,则

    $ {\rho _m}\left( {K,L} \right) \le {\vartheta _m}\left( {K,L} \right) \le \frac{{{\rho _K}}}{{{\rho _L}}} \le {\vartheta _M}\left( {K,L} \right) \le {\rho _M}\left( {K,L} \right) $ (4)

     ρm(K, L)和ρM(K, L)的定义[12]可分别等价于:

    $ \begin{array}{*{20}{c}} {{\rho _m}\left( {K,L} \right) = \frac{{{\rho _m}\left( {\partial K} \right)}}{{{\rho _M}\left( {\partial L} \right)}}}&{{\rho _M}\left( {K,L} \right) = \frac{{{\rho _M}\left( {\partial K} \right)}}{{{\rho _m}\left( {\partial L} \right)}}} \end{array} $

    ${{\vartheta }_{m}}$(K, L)与${{\vartheta }_{M}}$(K, L)的定义((3)式),不等式(4)显然成立.

    利用引理1,我们得到以下结果.

    定理1  设K, L为欧氏平面${{\mathbb{R}}^{2}}$中的卵形域,则:

    $ \left\{ \begin{array}{l} A_{K,L}^2 - {A_K}{A_L} \le {\left( {{A_{K,L}} - {\vartheta _m}\left( {K,L} \right){A_L}} \right)^2}\\ A_{K,L}^2 - {A_K}{A_L} \le {\left( {\frac{{{A_K}}}{{{\vartheta _m}\left( {K,L} \right)}} - {A_{K,L}}} \right)^2} \end{array} \right. $ (5)

    不等式(5)等号成立当且仅当KL位似.

     取(2)式中$C=-{{\vartheta }_{m}}\left(K, L \right)$,我们有

    $ {p_{{M_t}}} = \left( {1 - t} \right){p_K} - t{\vartheta _m}\left( {K,L} \right){p_L} $

    $0 < t\le \frac{1}{2}$时,由ρ的定义和引理1,得到

    $ {p_{{M_t}}} + \frac{{{\partial ^2}{p_{{M_t}}}}}{{\partial {\theta ^2}}} = \left( {1 - t} \right)\left( {{p_K} + {{p''}_K}} \right) - t{\vartheta _m}\left( {K,L} \right)\left( {{p_L} + {{p''}_L}} \right) \ge \left( {1 - t} \right){\rho _K} - t\frac{{{\rho _K}}}{{{\rho _L}}}{\rho _L} \ge 0 $

    因此${{p}_{{{M}_{t}}}}=\left(1-t \right){{p}_{K}}-t{{\vartheta }_{m}}\left(K, L \right){{p}_{L}}$是一类凸集Mt的支持函数,Mt的面积At

    $ \begin{array}{l} {A_t} = \frac{1}{2}\int_0^{2{\rm{ \mathsf{ π} }}} {{p_{{M_t}}}\left( {{p_{{M_t}}} + \frac{{{\partial ^2}{p_{{M_t}}}}}{{\partial {\theta ^2}}}} \right){\rm{d}}\theta } = \\ \;\;\;\;\;\;\;{\left( {1 - t} \right)^2}{A_K} - 2t\left( {1 - t} \right){\vartheta _m}\left( {K,L} \right){A_{K,L}} + {t^2}{\left( {{\vartheta _m}\left( {K,L} \right)} \right)^2}{A_L} \end{array} $

    特别地,当$t=\frac{1}{2}$时,有

    $ {A_{t = \frac{1}{2}}} = \frac{{{A_K}}}{4} - \frac{{{\vartheta _m}\left( {K,L} \right){A_{K,L}}}}{2} + \frac{{\vartheta _m^2\left( {K,L} \right){A_L}}}{4} \ge 0 $ (6)

    将(6)式改写成两种形式:

    $ - {A_K} \le \vartheta _m^2\left( {K,L} \right){A_L} - 2{\vartheta _m}\left( {K,L} \right){A_{K,L}} $
    $ - {A_L} \le \frac{{{A_K}}}{{\vartheta _m^2\left( {K,L} \right)}} - \frac{{2{A_{K,L}}}}{{{\vartheta _m}\left( {K,L} \right)}} $

    代入不等式(1)中,我们分别得到如下新的逆Bonnesen-型Minkowski不等式:

    $ A_{K,L}^2 - {A_K}{A_L} \le A_{K,L}^2 + \left( {\vartheta _m^2\left( {K,L} \right){A_L} - 2{\vartheta _m}\left( {K,L} \right){A_{K,L}}} \right){A_L} = {\left( {{A_{K,L}} - {\vartheta _m}\left( {K,L} \right){A_L}} \right)^2} $
    $ A_{K,L}^2 - {A_K}{A_L} \le A_{K,L}^2 + {A_K}\left( {\frac{{{A_K}}}{{\vartheta _m^2\left( {K,L} \right)}} - \frac{{2{A_{K,L}}}}{{{\vartheta _m}\left( {K,L} \right)}}} \right) = {\left( {\frac{{{A_K}}}{{{\vartheta _m}\left( {K,L} \right)}} - {A_{K,L}}} \right)^2} $

    不等式(5)等号成立的条件是${{M}_{t=\frac{1}{2}}}$的面积为0.此时${{M}_{t=\frac{1}{2}}}$只能是一条线段或一个点.如果${{M}_{t=\frac{1}{2}}}$是一条线段,则由(2)式可知,K的支持函数为

    $ {p_K} = 2{p_{{M_t}}} + {p_L}{\vartheta _m}\left( {K,L} \right) $

    这与K为卵形域矛盾,因此${{M}_{t=\frac{1}{2}}}$只能为一个点,即等号成立当且仅当KL位似.

    由定理1,我们得到以下逆Bonnesen-型Minkowski不等式:

    推论1  设K, L为欧氏平面${{\mathbb{R}}^{2}}$中的卵形域,则:

    $ \left\{ \begin{array}{l} A_{K,L}^2 - {A_K}{A_L} \le {\left( {{A_{K,L}} - {\rho _m}\left( {K,L} \right){A_L}} \right)^2}\\ A_{K,L}^2 - {A_K}{A_L} \le {\left( {\frac{{{A_K}}}{{{\rho _m}\left( {K,L} \right)}} - {A_{K,L}}} \right)^2} \end{array} \right. $ (7)

     注意到,定理1中(5)式不等号右边括号内几何量是非负的,即:

    $ \begin{array}{*{20}{c}} {{A_{K,L}} - {\vartheta _m}\left( {K,L} \right){A_L} \ge 0}&{\frac{{{A_K}}}{{{\vartheta _m}\left( {K,L} \right)}} - {A_{K,L}} \ge 0} \end{array} $ (8)

    由引理1,可证得

    $ {p_L}\left( {{p_K} + {{p''}_K}} \right) \ge {\vartheta _m}\left( {K,L} \right){p_L}\left( {{p_L} + {{p''}_L}} \right) $

    两边同时积分,我们有

    $ {A_{K,L}} = \frac{1}{2}\int_0^{2{\rm{ \mathsf{ π} }}} {{p_L}\left( {{p_K} + {{p''}_K}} \right){\rm{d}}\theta } \ge \frac{1}{2}{\vartheta _m}\left( {K,L} \right)\int_0^{2{\rm{ \mathsf{ π} }}} {{p_L}\left( {{p_L} + {{p''}_L}} \right){\rm{d}}\theta } = {\vartheta _m}\left( {K,L} \right){A_L} $

    因此,不等式(8)的第一个不等式成立.同理,可证得(8)式第二个不等式成立.由不等式(4)和(8),我们有:

    $ {A_{K,L}} - {\vartheta _m}\left( {K,L} \right){A_L} \le {A_{K,L}} - {\rho _m}\left( {K,L} \right){A_L} $
    $ \frac{{{A_K}}}{{{\vartheta _m}\left( {K,L} \right)}} - {A_{K,L}} \le \frac{{{A_K}}}{{{\rho _m}\left( {K,L} \right)}} - {A_{K,L}} $

    再由定理1可推出不等式(7).

    在文献[12]中,利用Blaschke滚动定理也得到了(7)式.由上述证明,可发现(5)式强于(7)式,因此我们加强了文献[12]中的结果.

    定理2  设K, L为欧氏平面${{\mathbb{R}}^{2}}$中的卵形域,则:

    $ \left\{ \begin{array}{l} A_{K,L}^2 - {A_K}{A_L} \le {\left( {{\vartheta _M}\left( {K,L} \right){A_L} - {A_{K,L}}} \right)^2}\\ A_{K,L}^2 - {A_K}{A_L} \le {\left( {{A_{K,L}} - \frac{{{A_K}}}{{{\vartheta _M}\left( {K,L} \right)}}} \right)^2} \end{array} \right. $ (9)

    不等式(9)等号成立当且仅当KL位似.

     用类似于定理1的讨论,取(2)式中$C=-{{\vartheta }_{M}}\left(K, L \right)$.当$\frac{1}{2}\le t < 1$时,我们得到一类凸集Mt,其支持函数为

    $ - {p_{{M_t}}} = t{\vartheta _M}\left( {K,L} \right){p_L} - \left( {1 - t} \right){p_K} $

    则其面积为

    $ {A_t} = {\left( {1 - t} \right)^2}{A_K} - 2t\left( {1 - t} \right){\vartheta _M}\left( {K,L} \right){A_{K,L}} + {t^2}{\left( {{\vartheta _M}\left( {K,L} \right)} \right)^2}{A_L} $

    特别地,当$t=\frac{1}{2}$时,有

    $ {A_{t = \frac{1}{2}}} = \frac{{{A_K}}}{4} - \frac{{{\vartheta _M}\left( {K,L} \right){A_{K,L}}}}{2} + \frac{{\vartheta _M^2\left( {K,L} \right){A_L}}}{4} \ge 0 $ (10)

    (10) 式可改写为:

    $ - {A_K} \le \vartheta _M^2\left( {K,L} \right){A_L} - 2{\vartheta _M}\left( {K,L} \right){A_{K,L}} $
    $ - {A_L} \le \frac{{{A_K}}}{{\vartheta _M^2\left( {K,L} \right)}} - \frac{{2{A_{K,L}}}}{{{\vartheta _M}\left( {K,L} \right)}} $

    代入不等式(1)中,得到不等式(9).同理,等号成立时,${{M}_{t=\frac{1}{2}}}$只能为一个点,即KL位似.

    类似推论1的证明,由定理2,可得以下较弱的逆Bonnesen-型Minkowski不等式.

    推论2  设K, L为欧氏平面${{\mathbb{R}}^{2}}$中的卵形域,则:

    $ \left\{ \begin{array}{l} A_{K,L}^2 - {A_K}{A_L} \le {\left( {{\rho _M}\left( {K,L} \right){A_L} - {A_{K,L}}} \right)^2}\\ A_{K,L}^2 - {A_K}{A_L} \le {\left( {{A_{K,L}} - \frac{{{A_K}}}{{{\rho _M}\left( {K,L} \right)}}} \right)^2} \end{array} \right. $ (11)

    由定理1和定理2,我们可得如下新的逆Bonnesen-型Minkowski不等式:

    定理3  设K, L为欧氏平面${{\mathbb{R}}^{2}}$中的卵形域,则

    $ A_{K,L}^2 - {A_K}{A_L} \le \frac{{A_L^2}}{4}{\left( {{\vartheta _M}\left( {K,L} \right) - {\vartheta _m}\left( {K,L} \right)} \right)^2} $ (12)

    等号成立当且仅当KL位似.

     将不等式(5),(9)中的第一个不等式,两边同时开根号,分别得到:

    $ \sqrt {A_{K,L}^2 - {A_K}{A_L}} \le {A_{K,L}} - {\vartheta _m}\left( {K,L} \right){A_L} $
    $ \sqrt {A_{K,L}^2 - {A_K}{A_L}} \le {\vartheta _m}\left( {K,L} \right){A_L} - {A_{K,L}} $

    不等式两边分别相加再平方,得到不等式(12).

    下面,我们将给出文献[12]中定理4.6等号成立条件.

    定理4  设K, L为欧氏平面${{\mathbb{R}}^{2}}$中的卵形域,则

    $ A_{K,L}^2 - {A_K}{A_L} \le \frac{{A_L^2}}{4}{\left( {{\rho _M}\left( {K,L} \right) - {\rho _m}\left( {K,L} \right)} \right)^2} $ (13)

    等号成立当且仅当KL为圆盘.

     由引理1和定理3,有

    $ A_{K,L}^2 - {A_K}{A_L} \le \frac{{A_L^2}}{4}{\left( {{\vartheta _M}\left( {K,L} \right) - {\vartheta _m}\left( {K,L} \right)} \right)^2} \le \frac{{A_L^2}}{4}{\left( {{\rho _M}\left( {K,L} \right) - {\rho _m}\left( {K,L} \right)} \right)^2} $ (14)

    下面证明(13)式等号成立.若K, L为圆盘,显然(13)式等号成立.反之,假设(13)式等号成立,则必有(14)式中第一个不等式等号成立,此时KL位似,从而

    $ A_{K,L}^2 - {A_K}{A_L} = \frac{{A_L^2}}{4}{\left( {{\vartheta _M}\left( {K,L} \right) - {\vartheta _m}\left( {K,L} \right)} \right)^2} = 0 $

    $ {\rho _M}\left( {K,L} \right) = {\rho _m}\left( {K,L} \right) $

    $ \frac{{{\rho _M}\left( {\partial K} \right)}}{{{\rho _m}\left( {\partial L} \right)}} = \frac{{{\rho _m}\left( {\partial K} \right)}}{{{\rho _M}\left( {\partial L} \right)}} $ (15)

    ρL不是常数,则有

    $ {\rho _M}\left( {\partial L} \right) > {\rho _m}\left( {\partial L} \right) $

    由于

    $ {\rho _M}\left( {\partial K} \right) \ge {\rho _m}\left( {\partial K} \right) $

    $ {\rho _M}\left( {K,L} \right) > {\rho _m}\left( {K,L} \right) $

    这与(15)式矛盾,故ρL是常数.再由(15)式,有

    $ {\rho _M}\left( {\partial K} \right) = {\rho _m}\left( {\partial K} \right) $

    ρK是常数.综上所述,等号成立当且仅当KL均为圆盘.

    参考文献
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    Reverse Bonnesen-Style Minkowski Inequalities in the Plane
    ZHOU Yuan1, ZHANG Zeng-le2     
    1. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China;
    2. School of Mathematics and Finance, Chongqing University of Arts and Sciences, Yongchuan Chongqing 402160, China
    Abstract: We study in this paper the upper bound of the Minkowski inequality in the plane, i.e. a reverse Bonnesen-style Minkowski inequality, such as AK, L2-AKALUK, L. Let K and L be convex bodies whose areas are AK and AL, respectively, and AK, L is the mixes area of the two convex bodies and UK, L is the geometric invariant related to K and L. We construct a class of convex body by the support function of the given convex bodies. By discussing the geometric properties of the new convex body, we obtain some new stronger reverse Bonnesen-style Minkowski inequalities and some results can be derived from those inequalities.
    Key words: convex body    support function    Minkowski inequality    reverse Bonnesen-style Minkowski inequality    
    X