带恢复公式与曲率恒等式的切片支撑函数
Slicing Support Functions with Recovery Formula and Curvature Identities
- National Yang Ming Chiao Tung University（国立阳明交通大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究推导了切片支撑函数的相关恒等式与恢复公式，建立了三维柱形Monge-Ampére型行列式恒等式，还推广了迭代切片支撑函数的构造并推导了全Hessian行列式恒等式。
AI中文摘要：
设$K\subset\mathbb{R}^n$为具有支撑函数$h_K$的凸体，对$\nu\in\mathbb{S}^{n-1}$、$p\in\mathbb{R}$及$u\in\nu^\perp$，我们引入切片支撑函数$h_\nu(u,p)$，定义为切片$K\cap\{x\cdot\nu=p\}$沿方向$u$的支撑函数。对每个固定的$p$，它恰好是Mathis与Meroni在构造凸纤维体时出现的对应平移纤维的支撑函数。我们推导了$h_\nu$基于$h_K$的下确界表示及对应的极小极大恒等式；利用Fenchel-Moreau定理证明，无需对$\partial K$作任何正则性假设，即可从切片支撑函数恢复$h_K$乃至$K$；当$K$严格凸且$\partial K$为$C^1$类时，还得到了微分恢复公式。在三维空间中，我们建立了以$\partial K$的球面曲率矩阵表示的柱形Monge-Ampére型行列式恒等式，当相关切方向为主方向时，该行列式简化为对应主曲率半径的加权比。我们进一步刻画了该主方向条件：对具有$C^2$边界且正高斯曲率的凸体，若除极点外的球面坐标方向均为主方向，则该体经平移后为旋转体。最后，我们将该构造推广到余维数更高的迭代切片支撑函数，并通过Schur补推导了全Hessian行列式恒等式。
英文摘要：
Let $K\subset\mathbb{R}^n$ be a convex body with support function $h_K$. For $ν\in\mathbb{S}^{n-1}$, $p\in\mathbb{R}$, and $u\inν^\perp$, we introduce the slicing support function $h_ν(u,p)$, defined as the support function of the slice $K\cap\{x\cdotν=p\}$ in the direction $u$. For each fixed $p$, this is precisely the support function of the corresponding translated fiber appearing in the construction of the convex fiber body of Mathis and Meroni \cite{MathisMeroni2023}. We derive an infimal representation of $h_ν$ in terms of $h_K$, together with a corresponding minimax identity. Using the Fenchel--Moreau theorem, we prove that $h_K$, and hence $K$, can be recovered from the slicing support function without any regularity assumption on $\partial K$. We also obtain a differential recovery formula when $K$ is strictly convex and $\partial K$ is of class $C^1$. In dimension three, we establish a cylindrical Monge--Ampére-type determinant identity expressed in terms of the spherical curvature matrix of $\partial K$. When the relevant tangent directions are principal directions, this determinant reduces to a weighted ratio of the corresponding principal radii of curvature. We further characterize this principal-direction condition by showing that, for convex bodies with $C^2$-boundary and positive Gaussian curvature, the spherical coordinate directions are principal directions away from the poles if and only if, up to translation, the body is a body of revolution. Finally, we extend the construction to higher-codimensional iterated slicing support functions and derive a full-Hessian determinant identity via the Schur complement.
