Slicing Support Functions with Recovery Formula and Curvature Identities

Abstract

Let $K\subset\mathbb{R}^n$ be a convex body with support function $h_K$. For $\nu\in\mathbb{S}^{n-1}$, $p\in\mathbb{R}$, and $u\in\nu^\perp$, we introduce the slicing support function $h_\nu(u,p)$, defined as the support function of the slice $K\cap\{x\cdot\nu=p\}$ 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 $h_\nu$ 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--Amp\'{e}re-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.