体积产品
Volume Product
摘要：
我们在这里的目的是给一个概述的已知结果和开放的问题，体积积${\mathcal P}(K)=\min_{z\in K}{\rm vol}(K){\rm vol}((K-z)^*)$的凸体$K$在${\mathbb R}^n$。本文给出了${\mathcal P}(K)$的一些上、下界，特别讨论了关于${\mathcal P}(K)$下界的Mahler猜想，该猜想至今尚未得到解决.我们还显示了${\mathcal P}(K)$与现代数学的不同部分的连接，包括几何数论，凸几何，分析，调和分析以及symmetry和辛几何和概率。
英文摘要：
Our purpose here is to give an overview of known results and open questions concerning the volume product ${\mathcal P}(K)=\min_{z\in K}{\rm vol}(K){\rm vol}((K-z)^*)$ of a convex body $K$ in ${\mathbb R}^n$. We present a number of upper and lower bounds for ${\mathcal P}(K)$, in particular, we discuss the Mahler's conjecture on the lower bound of ${\mathcal P}(K)$, which is still open. We also show connections of ${\mathcal P}(K)$ with different parts of modern mathematics, including Geometric Number Theory, Convex Geometry, Analysis, Harmonic Analysis as well as Systolic and Symplectic Geometries and Probability.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
