  Barker and Larman asked the following. Let $K' \subset {\Bbb{R}}^d$ be a
convex body, whose interior contains a given convex body $K \subset
{\Bbb{R}}^d$, and let, for all supporting hyperplanes $H$ of $K$, the
$(d-1)$-volumes of the intersections $K' \cap H$ be given. Is $K'$ 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 $K'$ 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 $K'$.
