论文标题
在$ L_P $ -BRUNN-MINKOWSKI和DIMENSION BRUNN-MINKOWSKI猜想中，用于log-conconcave措施
On the $L_p$-Brunn-Minkowski and dimensional Brunn-Minkowski conjectures for log-concave measures
论文作者
论文摘要
我们研究了有关对称性在Brunn-Minkowski类型不平等中的作用的最新猜想，例如Böröczky，Lutwak，Yang and Zhang和Zhang的$ L_P $ -Brunn-Minkowski猜想，以及Dimensional Brunn-Minkowski的Gardender of Gardure of Gardure of Gardure and Zvavitch in n ununive internner和Zvavitch。我们为这些猜想获得了几个新结果。 我们表明，当$ k \ subset l，$ $ l_p $ -brunn-minkowski的乘法形式适用于Lebesgue的量子，以$ p \ geq 1-cn^{ - 0.75} $的量子，该估计在某种情况下，它在某个身体中所包含的部分情况下，该估计会改善，从而改善了Kolesnikov和Milman的估计。 我们还表明，对于标准高斯度量的$ L_P $ -BRUNN-MINKOWSKI的乘法版本在包含足够大球的集合的情况下（其半径取决于$ P $）。特别是，当$ k $和$ l $包含$ \ sqrt {0.5（n+1）} b_2^n。 我们为对数洞穴度量的A a priori提出了更强的猜想，从而扩展了$ L_P $ -BRUNN-MINKOWSKI的猜想和尺寸的猜想，并在集合是扩张的情况下进行验证，并且该度量是高斯。我们还表明，log-brunn-minkowski的猜想，如果得到验证，将产生这种更普遍的不平等家庭。 我们的结果基于Kolesnikov和Milman以及Marsiglietti的Livshyts，Livshyts开发的方法。我们进一步验证了这些猜想的本地版本意味着一般度量设置的全局版本，并且此步骤使用Putterman最近开发的方法。
We study several of the recent conjectures in regards to the role of symmetry in the inequalities of Brunn-Minkowski type, such as the $L_p$-Brunn-Minkowski conjecture of Böröczky, Lutwak, Yang and Zhang, and the Dimensional Brunn-Minkowski conjecture of Gardner and Zvavitch, in a unified framework. We obtain several new results for these conjectures. We show that when $K\subset L,$ the multiplicative form of the $L_p$-Brunn-Minkowski conjecture holds for Lebesgue measure for $p\geq 1-Cn^{-0.75}$, which improves upon the estimate of Kolesnikov and Milman in the partial case when one body is contained in the other. We also show that the multiplicative version of the $L_p$-Brunn-Minkowski conjecture for the standard Gaussian measure holds in the case of sets containing sufficiently large ball (whose radius depends on $p$). In particular, the Gaussian Log-Brunn-Minkowski conjecture holds when $K$ and $L$ contain $\sqrt{0.5 (n+1)}B_2^n.$ We formulate an a-priori stronger conjecture for log-concave measures, extending both the $L_p$-Brunn-Minkowski conjecture and the Dimensional one, and verify it in the case when the sets are dilates and the measure is Gaussian. We also show that the Log-Brunn-Minkowski conjecture, if verified, would yield this more general family of inequalities. Our results build up on the methods developed by Kolesnikov and Milman as well as Colesanti, Livshyts, Marsiglietti. We furthermore verify that the local version of these conjectures implies the global version in the setting of general measures, and this step uses methods developed recently by Putterman.
