论文标题
关于Log-Brunn-Minkowski的本地版本，以及一些新的相关几何不等式
On the local version of the Log-Brunn-Minkowski conjecture and some new related geometric inequalities
论文作者
论文摘要
我们证明，对于任何半norm $ \ | \ cdot \ | $ on $ \ mathbb {r}^n，$和任何对称的convex body $ k $ in $ \ mathbb {r}^n，$ \ begin {equination} \frac{\|n_x\|^2}{\langle x,n_x\rangle}\leq \frac{1}{|K|}\left(\int_{\partial K} \|n_x\| \right)^2, \end{equation} and characterize the equality cases of this new inequality.如果证明了后者，以上也将遵循log-brunn-minkowski的猜想，并且我们认为这可能具有独立的利益。此外，在某些情况下，我们还可以改善这种不平等，涉及$ K的繁殖力。$ $。 在部分情况下，猜想的log-brunn-minkowski不平等现象是对对称凸的部分情况的加强，相当于以下陈述的有效性：对于所有对称的convex smooth $ k $ in $ \ mathbb {r}^n $ in Close clase flase flase flase flase clase flase clase flase clase flase clase flase clase clase flase compart \ Mathbb {r}，$ \ begin {qore} \ label {ineq-abs} \ int _ {\ partial k} h_x f^2- \ langle \ langle \ langle \ mbox {ii}^{ - 1}^{ - 1} +\ frac {f^2} {\ langle x，n_x \ rangle} \ leq \ frac {1} {| k |} \ left（\ int _ {\ int {\ partial k} f \ right）^2。 \ end {equation}在此注释中，我们使用速度函数的特定选择$ f（x）= | \ langle v，n_x \ rangle | $，对于所有对称的凸体$ k $，其中$ v \ in \ mathbb {r} n $是一个任意的向量。
We prove that for any semi-norm $\|\cdot\|$ on $\mathbb{R}^n,$ and any symmetric convex body $K$ in $\mathbb{R}^n,$ \begin{equation}\label{ineq-abs2} \int_{\partial K} \frac{\|n_x\|^2}{\langle x,n_x\rangle}\leq \frac{1}{|K|}\left(\int_{\partial K} \|n_x\| \right)^2, \end{equation} and characterize the equality cases of this new inequality. The above would also follow from the Log-Brunn-Minkowski conjecture, if the latter was proven, and we believe that it may be of independent interest. We, furthermore, obtain an improvement of this inequality in some cases, involving the Poincare constant of $K.$ The conjectured Log-Brunn-Minkowski inequality is a strengthening of the Brunn-Minkowski inequality in the partial case of symmetric convex bodies, equivalent to the validity of the following statement: for all symmetric convex smooth sets $K$ in $\mathbb{R}^n$ and all smooth even $f:\partial K\rightarrow \mathbb{R},$ \begin{equation}\label{ineq-abs} \int_{\partial K} H_x f^2-\langle \mbox{II}^{-1}\nabla_{\partial K} f, \nabla_{\partial K} f\rangle +\frac{f^2}{\langle x,n_x\rangle}\leq \frac{1}{|K|}\left(\int_{\partial K} f \right)^2. \end{equation} In this note, we verify the above with the particular choice of speed function $f(x)=|\langle v,n_x\rangle|$, for all symmetric convex bodies $K$, where $v\in\mathbb{R}^n$ is an arbitrary vector.
