论文标题
在多个$ l_p $ -curvilinear-brunn-minkowski不平等
On Multiple $L_p$-curvilinear-Brunn-Minkowski inequalities
论文作者
论文摘要
我们为有限的borel可测量集构建曲线求和的扩展，以$ l_p $ space用于多功率参数$ \barα=（α_1，\ cdots，α__{n+1}）$时$ p> 0 $。基于此$ l_ {p，\barα} $ - 集合的集合和概念集合的集合和概念，$ l_ {p，\barα} $ - curvilinear-brunn-minkowski不平等的不平等，用于界面可测量的bor borable-Borel可测量的集合和其正常化的版本。此外，通过利用虚拟图来实现功能，我们在$ l_ {p，\barα} $ borell-brascamp-lieb不平等及其归一化版本中制定了全新的证明，用于包含$ l_ {p} $ borell-brascamp-liebeality cy of the Persivation的功能$ l_ {p，\barα} $ - 集合的curvilinear-brunn-minkowski不等式。此外，我们提出了多个功率$ l_ {p，\barα} $ - 两个功能以及其属性以及其属性的上卷。 Last but not least, we introduce the definition of the surface area originated from the variation formula of measure in terms of the $L_{p,\barα}$-curvilinear summation for sets as well as $L_{p,\barα}$-supremal-convolution for functions together with their corresponding Minkowski type inequalities and isoperimetric inequalities for $p\geq1,$ ETC。
We construct the extension of the curvilinear summation for bounded Borel measurable sets to the $L_p$ space for multiple power parameter $\barα=(α_1, \cdots, α_{n+1})$ when $p>0$. Based on this $L_{p,\barα}$-curvilinear summation for sets and concept of {\it compression} of sets, the $L_{p,\barα}$-curvilinear-Brunn-Minkowski inequality for bounded Borel measurable sets and its normalized version are established. Furthermore, by utilizing the hypo-graphs for functions, we enact a brand new proof of $L_{p,\barα}$ Borell-Brascamp-Lieb inequality, as well as its normalized version, for functions containing the special case of $L_{p}$ Borell-Brascamp-Lieb inequality through the $L_{p,\barα}$-curvilinear-Brunn-Minkowski inequality for sets. Moreover, we propose the multiple power $L_{p,\barα}$-supremal-convolution for two functions together with its properties. Last but not least, we introduce the definition of the surface area originated from the variation formula of measure in terms of the $L_{p,\barα}$-curvilinear summation for sets as well as $L_{p,\barα}$-supremal-convolution for functions together with their corresponding Minkowski type inequalities and isoperimetric inequalities for $p\geq1,$ etc.
