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首页> 外文期刊>Differential geometry and its applications >Volume distance to hypersurfaces: Asymptotic behavior of its hessian
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Volume distance to hypersurfaces: Asymptotic behavior of its hessian

机译：到超曲面的体积距离：粗麻布的渐近行为

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摘要

The volume distance from a point p to a convex hypersurface M?RN+1 is defined as the minimum (N+1)-volume of a region bounded by M and a hyperplane H through the point. This function is differentiable in a neighborhood of M and if we restrict its hessian to the minimizing hyperplane H(p) we obtain, after normalization, a symmetric bilinear form Q. In this paper, we prove that Q converges to the affine Blaschke metric when we approximate the hypersurface along a curve whose points are centroids of parallel sections. We also show that the rate of this convergence is given by a bilinear form associated with the shape operator of M. These convergence results provide a geometric interpretation of the Blaschke metric and the shape operator in terms of the volume distance.
机译：从点p到凸超表面M≥RN+ 1的体积距离定义为由M和穿过该点的超平面H所界定的区域的最小（N + 1）体积。该函数在M的邻域上是可微的，如果将其粗略限制为最小化超平面H（p），则在归一化后，可以获得对称的双线性形式Q。在本文中，我们证明了Q收敛于仿射Blaschke度量，当我们沿一条曲线近似超曲面，该曲线的点是平行截面的质心。我们还表明，这种收敛的速率由与M的形状算符关联的双线性形式给出。这些收敛结果提供了Blaschke度量和形状算符在体积距离方面的几何解释。

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