论文标题
在$ L_P $ -BRUNN-MINKOWSKI类型和$ L_P $ -Isoperimetric类型的一般措施不平等现象
On $L_p$-Brunn-Minkowski type and $L_p$-isoperimetric type inequalities for general measures
论文作者
论文摘要
在2011年，lutwak，杨和张扩大了$ l_p $ -minkowski convex组合（$ p \ geq 1 $）的定义，这是Firey在1960年代在1960年代提出的，从凸面上包含其内部的凸起体，这些凸起的体内在其内部包含$ \ nathbb {r}^n $的所有可衡量子集，并将其与$ \ r}^n $，以及不平等（$ l_p $ -bmi）设置所有可测量的集合。在本文中，我们介绍了其$ L_P $ -Minkowski凸组合的功能扩展--- $ L_ {P，S} $ - 至上卷积，并证明$ L_P $ -BORELL-BRASCAMP-LIEB类型（$ L_P $ -BBL）的功能。基于$ L_P $ -BBL的功能类型不等式，我们将$ L_P $ -BMI用于可测量集的$ \ Mathbb {r}^n $具有$ \ weft（\ frac {1} {1} {s} {s} {s} {s} {s} {s} {s} {s} {s} {s} {s} {s} {s} {s} {s} {s} {s} {也就是说，我们表明，对于任何一对Borel设置$ a，b \ subset \ mathbb {r}^n $，任何$ t \ in [0,1] $和$ p \ geq 1 $，一个人具有\ [μ（（1-t） （1-t）μ（a）^{\ frac {p} {n+s}}+tμ（b）^{\ frac {\ frac {p} {n+s}}，\ \]，$μ$是$ \ mathbb {r}^n $ a $ \ weft a $ \ frac \ frac}的$ \ mathbb {r} $ 0 \ leq S <\ infty $。 Additionally, with the new defined $L_{p,s}$--supremal convolution for functions, we prove $L_p$-BMI for product measures with quasi-concave densities and for log-concave densities, $L_p$-Prékopa-Leindler type inequality ($L_p$-PLI) for product measures with quasi-concave densities, $ l_p $ -Minkowski的第一个不平等（$ L_P $ -MFI）和$ L_P $ iSoperimetric不平等（$ L_P $ -ISMI）用于一般措施等。最后，对Gardner-Zvavitch猜想的功能性对应物提出了$ P $ -P $ - 元整体化。
In 2011 Lutwak, Yang and Zhang extended the definition of the $L_p$-Minkowski convex combination ($p \geq 1$) introduced by Firey in the 1960s from convex bodies containing the origin in their interiors to all measurable subsets in $\mathbb{R}^n$, and as a consequence, extended the $L_p$-Brunn-Minkowski inequality ($L_p$-BMI) to the setting of all measurable sets. In this paper, we present a functional extension of their $L_p$-Minkowski convex combination---the $L_{p,s}$--supremal convolution and prove the $L_p$-Borell-Brascamp-Lieb type ($L_p$-BBL) inequalities. Based on the $L_p$-BBL type inequalities for functions, we extend the $L_p$-BMI for measurable sets to the class of Borel measures on $\mathbb{R}^n$ having $\left(\frac{1}{s}\right)$-concave densities, with $s \geq 0$; that is, we show that, for any pair of Borel sets $A,B \subset \mathbb{R}^n$, any $t \in [0,1]$ and $p\geq 1$, one has \[ μ((1-t) \cdot_p A +_p t \cdot_p B)^{\frac{p}{n+s}} \geq (1-t) μ(A)^{\frac{p}{n+s}} + t μ(B)^{\frac{p}{n+s}}, \] where $μ$ is a measure on $\mathbb{R}^n$ having a $\left(\frac{1}{s}\right)$-concave density for $0 \leq s < \infty$. Additionally, with the new defined $L_{p,s}$--supremal convolution for functions, we prove $L_p$-BMI for product measures with quasi-concave densities and for log-concave densities, $L_p$-Prékopa-Leindler type inequality ($L_p$-PLI) for product measures with quasi-concave densities, $L_p$-Minkowski's first inequality ($L_p$-MFI) and $L_p$ isoperimetric inequalities ($L_p$-ISMI) for general measures, etc. Finally a functional counterpart of the Gardner-Zvavitch conjecture is presented for the $p$-generalization.
