西南师范大学学报(自然科学版)   2019, Vol. 44 Issue (12): 31-34.  DOI: 10.13718/j.cnki.xsxb.2019.12.006
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  • 对偶Lq变换法则    [PDF全文]
    陶江艳 , 李晓     
    西南大学 数学与统计学院, 重庆 400715
    摘要:在已有结果的基础上对对偶Lq Brunn-Minkowski理论做了一些推广,主要讨论了对偶Lq Brunn-Minkowski型不等式并得到部分结果.给出统一的处理对偶Lq Brunn-Minkowski型不等式的方法,称此方法为对偶Lq变换法则,通过运用此法则给出了关于对偶混合体积著名的对偶Lq Brunn-Minkowski型不等式的简化证明.
    关键词对偶混合体积    对偶Brunn-Minkowski型不等式    对偶Lq Brunn-Minkowski型不等式    

    Brunn-Minkowski不等式是经典的等周不等式的推广,也是Brunn-Minkowski理论中重要的不等式之一[1-3].对偶Brunn-Minkowski理论是经典Brunn-Minkowski理论的自然发展.对偶Brunn-Minkowski不等式是对偶Brunn-Minkowski理论中最重要的不等式之一[4-6].

    KL是$\mathbb { R } ^ { n }$中关于原点的星体,则有

    $ V{(K\tilde + L)^{\frac{1}{n}}} \leqslant V{(K)^{\frac{1}{n}}} + V{(L)^{\frac{1}{n}}} $ (1)

    等号成立当且仅当KL互为膨胀,其中V(·)是n维的体积,$K\tilde + L$表示KL的径向和.

    20世纪60年代初,文献[7]介绍了凸体的Lq加法及数乘,并建立了Lq Brunn-Minkowski不等式.文献[8-9]推动了Lq Brunn-Minkowski理论的进一步发展.关于Lq Brunn-Minkowski理论以及对偶Lq Brunn-Minkowski理论的最新讨论参见文献[10-12].对偶Lq Brunn-Minkowski不等式为:

    KL是$\mathbb { R } ^ { n }$中关于原点的星体,0 < q < n,则有

    $ V{\left( {K{{\tilde + }_q}L} \right)^{\frac{q}{n}}} \leqslant V{(K)^{\frac{q}{n}}} + V{(L)^{\frac{q}{n}}} $ (2)

    等号成立当且仅当KL互为膨胀,其中$K{\tilde + _q}L$表示KLq阶径向和.

    本文主要研究对偶Brunn-Minkowski型不等式,用对偶Lq变换法则证明了与对偶混合体积有关的不等式.

    有关凸几何的基本知识和常用符号可参见文献[3].

    定义1   给定函数:$\tilde F:S_o^n \to (0, + \infty), K, L \in S_o^n$,若对α∈(0,1),有

    $ \tilde F{\left( {(1 - \alpha ){ \cdot _q}K{{\tilde + }_q}\alpha { \cdot _q}L} \right)^q} \leqslant (1 - \alpha )\tilde F{(K)^q} + \alpha \tilde F{(L)^q} $

    则称$\tilde F$是q-凸的.当q=1时,我们称$\tilde F$为凸的.

    引理1   设KLSon,0 < q < 1,0 < α < 1,则

    $ (1 - \alpha ){ \cdot _q}K{{\tilde + }_q}\alpha { \cdot _q}L \subseteq (1 - \alpha )K\tilde + \alpha L $

    等号成立当且仅当K=L.

      由$(1 - \alpha){ \cdot _q}K{\tilde + _q}\alpha { \cdot _q}L$的定义和0 < q < 1,t∈(0,∞)时f(t)=tq的严格凹性,对所有的uSn-1,有

    $ \begin{gathered} \rho {\left( {(1 - \alpha ){ \cdot _q}K{{\tilde + }_q}\alpha { \cdot _q}L,u} \right)^q} = (1 - \alpha )\rho {(K,u)^q} + \alpha \rho {(L,u)^q} \leqslant \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{((1 - \alpha )\rho (K,u) + \alpha \rho (L,u))^q} = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\rho {((1 - \alpha )K\tilde + \alpha L,u)^q} \hfill \\ \end{gathered} $

    所以

    $ (1 - \alpha ){ \cdot _q}K{{\tilde + }_q}\alpha { \cdot _q}L \subseteq (1 - \alpha )K\tilde + \alpha L $

    等号成立当且仅当

    $ \rho (K,u) = \rho (L,u)\;\;\;\forall u \in {S^{n - 1}} $

    即当且仅当K=L.

    定理1   设:$\tilde F:S_o^n \to (0, + \infty)$是正齐次的、增的凸函数,0 < q < 1.设KLSon,则对所有的α∈(0,1),有

    $ \tilde F{\left( {(1 - \alpha ){ \cdot _q}K{{\tilde + }_q}\alpha { \cdot _q}L} \right)^q} \leqslant (1 - \alpha )\tilde F{(K)^q} + \alpha F{(L)^q} $ (3)

    当:$\tilde F:S_o^n \to (0, + \infty)$为严格增时,(3) 式中的等号成立当且仅当KL互为膨胀.

      不等式(3)等价于

    $ \frac{{\tilde F\left( {(1 - \alpha ){ \cdot _q}K{{\tilde + }_q}\alpha { \cdot _q}L} \right)}}{{{{\left( {(1 - \alpha )\tilde F{{(K)}^q} + \alpha \tilde F{{(L)}^q}} \right)}^{\frac{1}{q}}}}} \leqslant 1 $ (4)

    由$\tilde F$的正齐次性,(4)式等价于

    $ \tilde F\left( {\frac{{(1 - \alpha ){ \cdot _q}K{{\tilde + }_q}\alpha { \cdot _q}L}}{{{{\left( {(1 - \alpha )\tilde F{{(K)}^q} + \alpha \tilde F{{(L)}^q}} \right)}^{\frac{1}{q}}}}}} \right) \leqslant 1 $

    $ \tilde F\left( {\left( {\frac{{1 - \alpha }}{{(1 - \alpha )\tilde F{{(K)}^q} + \alpha \tilde F{{(L)}^q}}}} \right){ \cdot _q}K{{\tilde + }_q}\left( {\frac{\alpha }{{(1 - \alpha )\tilde F{{(K)}^q} + \alpha \tilde F{{(L)}^q}}}} \right){ \cdot _q}L} \right) \leqslant 1 $ (5)

    又因

    $ \begin{gathered} \left( {\frac{{1 - \alpha }}{{(1 - \alpha )\tilde F{{(K)}^q} + \alpha \tilde F{{(L)}^q}}}} \right){ \cdot _q}K{{\tilde + }_q} \\\left( {\frac{\alpha }{{(1 - \alpha )\tilde F{{(K)}^q} + \alpha \tilde F{{(L)}^q}}}} \right){ \cdot _q}L = \hfill \\ \left( {\frac{{\left( {1 - \alpha } \right)\tilde F{{\left( K \right)}^q}}}{{(1 - \alpha )\tilde F{{(K)}^q} + \alpha \tilde F{{(L)}^q}}}} \right) \\{ \cdot _q}\left( {\frac{K}{{\tilde F\left( K \right)}}} \right){{\tilde + }_q}\left( {\frac{{\alpha \tilde F{{\left( L \right)}^q}}}{{(1 - \alpha )\tilde F{{(K)}^q} + \alpha \tilde F{{(L)}^q}}}} \right){ \cdot _q}\left( {\frac{L}{{\tilde F\left( L \right)}}} \right) = \hfill \\ \left( {1 - \alpha '} \right){ \cdot _q}\left( {\frac{K}{{\tilde F\left( K \right)}}} \right){{\tilde + }_q}\alpha '{ \cdot _q}\left( {\frac{L}{{\tilde F\left( L \right)}}} \right) \hfill \\ \end{gathered} $ (6)

    其中

    $ \alpha ' = \frac{{\alpha \tilde F{{\left( L \right)}^q}}}{{(1 - \alpha )\tilde F{{(K)}^q} + \alpha \tilde F{{(L)}^q}}} $

    结合(5)式,不等式(3)等价于

    $ \tilde F\left( {\left( {1 - \alpha '} \right){ \cdot _q}\left( {\frac{K}{{\tilde F\left( K \right)}}} \right){{\tilde + }_q}\alpha '{ \cdot _q}\left( {\frac{L}{{\tilde F\left( L \right)}}} \right)} \right) \leqslant 1 $

    对(6)式运用引理1,得

    $ \left( {1 - \alpha '} \right){ \cdot _q}\left( {\frac{K}{{\tilde F\left( K \right)}}} \right){{\tilde + }_q}\alpha '{ \cdot _q}\left( {\frac{L}{{\tilde F\left( L \right)}}} \right) \subseteq \left( {1 - \alpha '} \right)\left( {\frac{K}{{\tilde F\left( K \right)}}} \right)\tilde + \alpha '\left( {\frac{L}{{\tilde F\left( L \right)}}} \right) $

    因此,由$\tilde F$的单调性、凸性和正齐次性,有

    $ \begin{gathered} \tilde F\left( {\left( {1 - \alpha '} \right){ \cdot _q}\left( {\frac{K}{{\tilde F\left( K \right)}}} \right){{\tilde + }_q}\alpha '{ \cdot _q}\left( {\frac{L}{{\tilde F\left( L \right)}}} \right)} \right) \leqslant \hfill \\ \tilde F\left( {\left( {1 - \alpha '} \right)\left( {\frac{K}{{\tilde F\left( K \right)}}} \right)\tilde + \alpha '\left( {\frac{L}{{\tilde F\left( L \right)}}} \right)} \right) \leqslant \hfill \\ \left( {1 - \alpha '} \right)\left( {\frac{K}{{\tilde F\left( K \right)}}} \right)\tilde + \alpha '\left( {\frac{L}{{\tilde F\left( L \right)}}} \right) = \hfill \\ \left( {1 - \alpha '} \right) + \alpha ' = 1 \hfill \\ \end{gathered} $ (7)

    假设(3)式中等号成立,则(7)式中不等号应为等号,即

    $ \tilde F\left( {\left( {1 - \alpha '} \right){ \cdot _q}\left( {\frac{K}{{\tilde F\left( K \right)}}} \right){{\tilde + }_q}\alpha '{ \cdot _q}\left( {\frac{L}{{\tilde F\left( L \right)}}} \right)} \right) = \tilde F\left( {\left( {1 - \alpha '} \right)\left( {\frac{K}{{\tilde F\left( K \right)}}} \right)\tilde + \alpha '\left( {\frac{L}{{\tilde F\left( L \right)}}} \right)} \right) $

    因$\tilde F$是严格增函数,则

    $ \left( {1 - \alpha '} \right){ \cdot _q}\left( {\frac{K}{{\tilde F\left( K \right)}}} \right){{\tilde + }_q}\alpha '{ \cdot _q}\left( {\frac{L}{{\tilde F\left( L \right)}}} \right) = \left( {1 - \alpha '} \right)\left( {\frac{K}{{\tilde F\left( K \right)}}} \right)\tilde + \alpha '\left( {\frac{L}{{\tilde F\left( L \right)}}} \right) $

    又由引理1知

    $ \frac{K}{{\tilde F\left( K \right)}} = \frac{L}{{\tilde F\left( L \right)}} $

    所以KL互为膨胀.

    另一方面,假设KL互为膨胀,即K=βLβ>0,则有

    $ \begin{gathered} (1 - \alpha ){ \cdot _q}K{{\tilde + }_q}\alpha { \cdot _q}L = (1 - \alpha ){ \cdot _q}\left( {\beta L} \right){{\tilde + }_q}\alpha { \cdot _q}L = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{(1 - \alpha )^{\frac{1}{q}}}\left( {\beta L} \right){{\tilde + }_q}{\alpha ^{\frac{1}{q}}}L = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left( {\left( {1 - \alpha } \right){\beta ^q} + \alpha } \right)^{\frac{1}{q}}}L \hfill \\ \end{gathered} $

    由$\tilde F$的正齐次性得到

    $ \begin{gathered} \tilde F{\left( {(1 - \alpha ){ \cdot _q}K{{\tilde + }_q}\alpha { \cdot _q}L} \right)^q} = \tilde F{\left( {{{\left( {\left( {1 - \alpha } \right){\beta ^q} + \alpha } \right)}^{\frac{1}{q}}}L} \right)^q} = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {1 - \alpha } \right){\beta ^q}\tilde F{\left( L \right)^q} + \alpha \tilde F{\left( L \right)^q} = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {1 - \alpha } \right)\tilde F{\left( K \right)^q} + \alpha \tilde F{\left( L \right)^q} \hfill \\ \end{gathered} $

    即(3)式中等号成立.

    推论1   设:$\tilde F:S_o^n \to (0, + \infty)$是正齐次的、增的凸函数,0 < q < 1.设KLS on,则

    $ \tilde F{\left( {K{{\tilde + }_q}L} \right)^q} \leqslant \tilde F{\left( K \right)^q} + \tilde F{\left( L \right)^q} $ (8)

    当:$\tilde F:S_o^n \to (0, + \infty)$为严格增时,(8)式中等号成立当且仅当KL互为膨胀.

      设α ∈(0,1),由定理1和$\tilde F$的正齐次性,得到

    $ \begin{gathered} \tilde F{\left( {K{{\tilde + }_q}L} \right)^q} = \tilde F{\left( {\left( {1 - \alpha } \right){ \cdot _q}\left( {{{\left( {1 - \alpha } \right)}^{ - \frac{1}{q}}}K} \right)\tilde + {}_q\alpha { \cdot _q}\left( {{\alpha ^{ - \frac{1}{q}}}} \right)L} \right)^q} \leqslant \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {1 - \alpha } \right)\tilde F{\left( {{{\left( {1 - \alpha } \right)}^{ - \frac{1}{q}}}K} \right)^q} + \alpha \tilde F{\left( {{\alpha ^{ - \frac{1}{q}}}L} \right)^q} = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\tilde F{\left( K \right)^q} + \tilde F{\left( L \right)^q} \hfill \\ \end{gathered} $

    当$\tilde F$为严格增时,等号成立当且仅当${(1 - \alpha)^{ - \frac{1}{q}}}K$与${\alpha ^{ - \frac{1}{q}}}L$互为膨胀,所以KL互为膨胀.

    下面给出著名的对偶Brunn-Minkowski型不等式[3]的一个简化证明:

    定理2  设KLKj+1,…,KnSon,0 < q < 1且j∈{2,…,n},则

    $ \tilde V{\left( {K{{\tilde + }_q}L,j;{K_{j + 1}}, \cdots ,{K_n}} \right)^{\frac{a}{j}}} \leqslant \tilde V{\left( {K,j;{K_{j + 1}}, \cdots ,{K_n}} \right)^{\frac{a}{j}}} + \tilde V{\left( {L,j;{K_{j + 1}}, \cdots ,{K_n}} \right)^{\frac{q}{j}}} $

    等号成立当且仅当KL互为膨胀.

      设KLKj+1,…,KnSon

    $ \tilde F\left( K \right) = \tilde V{\left( {K,j;{K_{j + 1}}, \cdots ,{K_n}} \right)^{\frac{1}{j}}} $

    则$\tilde F$是正齐次的、严格增的正函数.由文献[3]知$\tilde F$是凸函数.所以由推论1知

    $ \tilde F{\left( {K{{\tilde + }_q}L} \right)^q} \leqslant \tilde F{\left( K \right)^q} + \tilde F{\left( L \right)^q} $

    $ \tilde V{\left( {K{{\tilde + }_q}L,j;{K_{j + 1}}, \cdots ,{K_n}} \right)^{\frac{a}{j}}} \leqslant \tilde V{\left( {K,j;{K_{j + 1}}, \cdots ,{K_n}} \right)^{\frac{a}{j}}} + \tilde V{\left( {L,j;{K_{j + 1}}, \cdots ,{K_n}} \right)^{\frac{q}{j}}} $

    等号成立当且仅当KL互为膨胀.

    参考文献
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    杨琴, 罗淼. 平面上几个对偶Brunn-Minkowski不等式[J]. 西南大学学报(自然科学版), 2015, 37(2): 74-78.
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    LUTWAKE. The Brunn-Minkowski-Firey Theory II[J]. Advance in Mathematics, 1996, 118(2): 244-294.
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    ZOU D, XIONG G. A Unified Treatment for Lp Brunn-Minkowski Type Inequalities[J]. Comm Anal Geom, 2018, 26(2): 435-460. DOI:10.4310/CAG.2018.v26.n2.a6
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    杨林, 罗淼, 王贺军. Lp对偶Brunn-Minkowski不等式[J]. 西南大学学报(自然科学版), 2017, 39(10): 79-83.
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    杨林, 罗淼, 吴笑雪, 等. Lp对偶Minkowski不等式的稳定性[J]. 西南师范大学学报(自然科学版), 2018, 43(4): 37-40.
    Dual Lq Transference Principle
    TAO Jiang-yan , LI Xiao     
    School of Mathematics and Statistics, SouthwestUniversity, Chongqing 400715, China
    Abstract: In this paper, the dual Lq Brunn-Minowski theory is generalized based on the existing results and mainly discusses the dual Lq Brunn-Minowski type inequality and obtains few results. In this paper, a unified method for dealing with Lq Brunn-Minowski type inequality is given by means of literature review, which is called dual Lq transference principle. By using this principle, a simplified proof of the famous dual Lq Brunn-Minkowski type inequality about the dual mixed volume is given.
    Key words: dual mixed volume    dual Brunn-Minkowski type inequalities    dual Lq Brunn-Minkowski type inequaliyies    
    X