4 problems

Let $K\subset\mathbb R^n$, $n\geq 3$, be a strictly convex body, and let $\mathcal S$ be a family of hyperplanes that is the image of an embedding of the tangent bundle of the sphe…

Let $K,S\subset\mathbb R^n$, $n\geq 3$, be convex bodies with $S\subset\operatorname{int}K$. Barker–Larman conjecture. If, for every hyperplane $H$ supporting $S$, the section…

Let $K\subset\mathbb R^n$ be a convex body with $n\geq 3$, and let $x\in\mathbb R^n$. Suppose that every $2$-dimensional section of $K$ through $x$ has a centre of symmetry. Rogers…

Let $K$ and $L$ be convex bodies in $\mathbb R^n$, with $n\geq 3$, and suppose that … For every pair of parallel support hyperplanes of $K$, consider their sections with $L$ and th…