论文标题
期望随机亚曼叶片：圆锥形截面
Expectation of a random submanifold: the zonoid section
论文作者
论文摘要
我们开发基于Zonoids（一种特殊类别的凸体）的演算，以期与与随机submanifold $ z $相关的功能，定义为Riemannian歧管上的平滑矢量值随机场的零集。我们在随机字段上确定了一组方便的假设，在该字段下定义了其Zonoid截面，在歧管的每个点$ p $的cotangengent空间的外部代数中分配了Zonoid $ζ（P）$。我们证明，$ζ（p）$的第一个内在体积是$ z $的预期量的kac-rice密度，而其中心计算超过$ z $的预期集成电流。我们表明，随机亚策略的相交对应于Zonoid切片的楔形产物，并且预映射对应于下拉背包。 将其与最近开发的宗教代数相结合，它允许为KAC-RICE公式提供乘法结构，类似于歧管的共同体学环。此外，它与凸体和估值理论建立了联系，其中包括非常深刻而困难的结果，例如Alexandrov-Fenchel不平等和Brunn-Minkowsky的不平等。我们将它们导出到这种情况下，以证明随机子延伸物的两个类似的新不平等现象。在Finsler几何形状的背景下应用我们的结果，我们证明了一些新的Crofton公式，用于曲线的长度和Finsler歧管中的Submanifolds的Holmes-Thompson卷。
We develop a calculus based on zonoids - a special class of convex bodies - for the expectation of functionals related to a random submanifold $Z$ defined as the zero set of a smooth vector valued random field on a Riemannian manifold. We identify a convenient set of hypotheses on the random field under which we define its zonoid section, an assignment of a zonoid $ζ(p)$ in the exterior algebra of the cotangent space at each point $p$ of the manifold. We prove that the first intrinsic volume of $ζ(p)$ is the Kac-Rice density of the expected volume of $Z$, while its center computes the expected current of integration over $Z$. We show that the intersection of random submanifolds corresponds to the wedge product of the zonoid sections and that the preimage corresponds to the pull-back. Combining this with the recently developed zonoid algebra, it allows to give a multiplication structure to the Kac-Rice formulas, resembling that of the cohomology ring of a manifold. Moreover, it establishes a connection with the theory of convex bodies and valuations, which includes very deep and difficult results such as the Alexandrov-Fenchel inequality and the Brunn-Minkowsky inequality. We export them to this context to prove two analogous new inequalities for random submanifolds. Applying our results in the context of Finsler geometry, we prove some new Crofton formulas for the length of curves and the Holmes-Thompson volumes of submanifolds in a Finsler manifold.
