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 \(p\ge 1-Cn^{-0.75}\), 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.

我们研究了关于对称在Brunn–Minkowski型不等式中的作用的一些最新猜想，例如Böröczky，Lutwak，Yang和Zhang的\（L_p \）- Brunn–Minkowski猜想以及尺寸Brunn–在统一框架中，Gardner和Zvavitch的Minkowski猜想。对于这些猜想，我们获得了一些新的结果。我们证明，当\（K \ subset L，\）\（L_p \）- Brunn–Minkowski猜想的乘法形式适用于\（p \ ge 1-Cn ^ { -0.75} \）的Lebesgue测度时，当一个物体包含在另一个物体中时，在部分情况下改进了Kolesnikov和Milman的估计。我们还表明\（L_p \）的乘法形式对于包含足够大的球（其半径取决于p）的集合，标准高斯测度的-Brunn-Minkowski猜想成立。特别地，所述高斯日志-Brunn的-闵可夫斯基猜想成立时ķ和大号包含\（\ SQRT {0.5（N + 1）} B_2 ^ N \）我们制定数凹措施先验更强猜想，延伸两者\（L_p \）-Brunn–Minkowski猜想和Dimensional猜想，并在集合为扩张且度量为高斯的情况下进行验证。我们还表明，如果验证了Log-Brunn-Minkowski猜想，将得出这个更一般的不等式族。我们的结果建立在Kolesnikov和Milman以及Colesanti，Livshyts和Marsiglietti开发的方法的基础上。我们进一步验证了这些猜想的局部版本暗示了通用度量设置中的全局版本，并且此步骤使用了Putterman最近开发的方法。