论文标题
敏锐的等级不平等现象
Sharp Isoperimetric Inequalities for Affine Quermassintegrals
论文作者
论文摘要
$ \ mathbb {r}^n $中与凸体相关的仿生QuermassIntegrals是Brunn-Minkowski理论的经典固有体积的仿射不变的类似物，因此构成了仿射凸线的中心支柱。他们是由E. Lutwak在1980年代引入的，E. Lutwak猜想，在给定量的所有凸面中，$ k $ th的Aggine QuermassIntegral被精确地最小化了椭圆形的家族。已知案例$ k = 1 $和$ k = n-1 $分别对应于经典的blaschke-santaló和小投影不平等。在这项工作中，我们确认了卢特瓦克的猜想，包括对平等案例的表征，对于$ k = 1，\ ldots，n-1 $，在一个单一的统一框架中。实际上，事实证明，相对于Hausdorff拓扑，椭圆形是唯一的本地最小化器。 为了证明证明，我们引入了许多新成分，包括凸体的投影Rolodex的新颖结构。特别是，从这个新的角度来看，Petty的不平等被解释为用于由投影Rolodex编码的新的极性体系列的广义blaschke-Santaló不平等的综合形式。我们将这些结果扩展到更通用的$ l^p $ - 大小的QuermassIntegrals，并将其解释为$ p = 0 $的情况，为平均的Loomis-Whitney Isoperimetric不平等。
The affine quermassintegrals associated to a convex body in $\mathbb{R}^n$ are affine-invariant analogues of the classical intrinsic volumes from the Brunn-Minkowski theory, and thus constitute a central pillar of affine convex geometry. They were introduced in the 1980's by E. Lutwak, who conjectured that among all convex bodies of a given volume, the $k$-th affine quermassintegral is minimized precisely on the family of ellipsoids. The known cases $k=1$ and $k=n-1$ correspond to the classical Blaschke-Santaló and Petty projection inequalities, respectively. In this work we confirm Lutwak's conjecture, including characterization of the equality cases, for all values of $k=1,\ldots,n-1$, in a single unified framework. In fact, it turns out that ellipsoids are the only local minimizers with respect to the Hausdorff topology. For the proof, we introduce a number of new ingredients, including a novel construction of the Projection Rolodex of a convex body. In particular, from this new view point, Petty's inequality is interpreted as an integrated form of a generalized Blaschke--Santaló inequality for a new family of polar bodies encoded by the Projection Rolodex. We extend these results to more general $L^p$-moment quermassintegrals, and interpret the case $p=0$ as a sharp averaged Loomis--Whitney isoperimetric inequality.
