Unique local determination of convex bodies

Abstract

Barker and Larman asked the following. Let $K^{\prime }\subset {\mathbb{R}}^d$ be a convex body, whose interior contains a given convex body $K\subset {\mathbb{R}}^d$, and let, for all supporting hyperplanes $H$ of $K$, the $(d-1)$-volumes of the intersections $K^{\prime }\cap H$ be given. Is $K^{\prime }$ then uniquely determined? Yaskin and Zhang asked the analogous question when, for all supporting hyperplanes $H$ of $K$, the $d$-volumes of the "caps" cut off from $K^{\prime }$ by $H$ are given. We give local positive answers to both of these questions, for small $C^2$-perturbations of $K$, provided the boundary of $K$ is $C_{+}^2$. In both cases, $(d-1)$-volumes or $d$-volumes can be replaced by $k$-dimensional quermassintegrals for $1\le k\le d-1$ or for $1\le k\le d$, respectively. Moreover, in the first case we can admit, rather than hyperplane sections, sections by $l$-dimensional affine planes, where $1\le k\le l\le d-1$. In fact, here not all $l$-dimensional affine subspaces are needed, but only a small subset of them (actually, a $(d-1)$-manifold), for unique local determination of $K^{\prime }$.

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