On counterexamples in questions of unique determination of convex bodies (Q2839371)

The authors discuss a certain construction that allows to give counterexamples to many problems on unique determination of convex bodies. In particular, this construction is applied to the following problem from the book ``Geometric tomography 2nd ed.'' by \textit{R. J. Gardner} [Cambridge: Cambridge University Press (2006; Zbl 1102.52002)]. Let \(K\) and \(L\) be origin-symmetric convex bodies in \(\mathbb R^3\) such that the sections \(K \cap H\) and \(L\cap H\) have equal perimeters for every two-dimensional subspace \(H\) of \(\mathbb R^3\). Does it follow that \(K=L\)? The authors show that without the symmetry assumption the question has a negative answer. The latter is shown in full generality for convex bodies in \(\mathbb R^n\) and any intrinsic volumes instead of perimeters. The authors also pose a problem about unique determination of convex bodies by slabs. For this problem, uniqueness also fails in the absence of the symmetry assumption.