一个具有中心对称截面的板条中的尖锐Santaló不等式
A sharp Santaló inequality in a slab with centrally symmetric sections
- School of Mathematical Sciences, Tel Aviv University（特拉维夫大学数学科学学院）
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中文总结 AI 辅助
本文证明了一个尖锐的Santaló型不等式，通过射影对应转化为板条中凸体的体积乘积上界问题，并分类了极值情形，同时给出了一维函数不等式族。
中文摘要 AI 辅助
我们证明了关于$V(\varphi)$与$\varphi$的Legendre变换的$V$之积的一个尖锐的Santaló型不等式，其中$\varphi$是$\mathbb{R}^n$上的偶几何凸函数，且$V(\varphi)=\int(1+\varphi)^{-(n+1)}$。对于这个尺寸泛函，Legendre变换与函数极性给出相同的体积，因此该不等式对两种对偶性均成立。通过射影对应，该问题等价于关于标准体积和标准极性的一类未必中心对称的凸体的新的Santaló型不等式。该类由包含$\{0\}\times[-1,1]$的凸体$K\subset\mathbb R^n\times[-1,1]$组成，其中每个水平截面关于指定的垂直轴是中心对称的。我们确定了该类中$\operatorname{vol}_{n+1}(K)\operatorname{vol}_{n+1}(K^\circ)$的尖锐上界，并分类了所有等号情形。当$n=1$时，极值体是圆盘及其水平线性像。当$n\ge 2$时，球体不再是极大值点，极值体在水平线性变换和反射$t\mapsto -t$意义下唯一。应用于水平截面的对称Santaló不等式与单调输运一起，将该问题归结为正方形$[-1,1]^2$中递增曲线的极大化问题。我们利用$n=1$时的全局势和$n\ge 2$时的两个Hamilton-Jacobi分支解决了该问题。最后，在一维情形下，我们提供了Legendre对偶性的尖锐函数Santaló不等式族，在体积$V$与指数体积$\int \exp(-\varphi)$之间插值，且以中心二次型为唯一等号情形。
英文摘要
We prove a sharp Santaló-type inequality for the product of $V(φ)$ and $V$ of the Legendre transform of $φ$, where $φ$ is an even geometric convex function on $\mathbb{R}^n$ and $V(φ)=\int(1+φ)^{-(n+1)}$. For this size functional, the Legendre transform and functional polarity give the same volume, so the same inequality holds for both dualities. Through a projective correspondence, the problem is equivalent to a new Santaló-type inequality with respect to standard volume and standard polarity, on a class of not necessarily centered convex bodies. This class consists of convex bodies $K\subset\mathbb R^n\times[-1,1]$ containing $\{0\}\times[-1,1]$, for which every horizontal section is centrally symmetric about the distinguished vertical axis. We determine the sharp upper bound for $\operatorname{vol}_{n+1}(K)\operatorname{vol}_{n+1}(K^\circ)$ in this class and classify all equality cases. When $n=1$, the extremizers are the disk and its horizontal linear images. When $n\ge 2$, the ball is no longer a maximizer, and the extremizer is unique up to horizontal linear transformations and the reflection $t\mapsto -t$. The symmetric Santaló inequality applied to horizontal sections, together with monotone transport, reduces the problem to a maximization problem for increasing curves in the square $[-1,1]^2$. We solve this problem using a global potential when $n=1$ and two Hamilton-Jacobi branches when $n\ge 2$. Finally, in dimension one we provide a sharp family of functional Santaló inequalities for the Legendre duality, interpolating between the volume $V$ and the exponential volume $\int \exp(-φ)$, with centered quadratics as the only equality cases.
