4 problems

Let $K,S\subset\mathbb R^n$, $n\geq 3$, be convex bodies with $S\subset\operatorname{int}K$. Inversely homothetic sections conjecture. If, for every pair of parallel hyperplanes…

Central-slice extremality conjecture. For every centrally symmetric $k$-dimensional polytope $P$, the upper bound on the number of vertices of a $(k-1)$-dimensional slice of $P$ is…

Let $K,L\subset\mathbb R^2$ be convex bodies with $L\subset\operatorname{int}K$, and suppose that $L$ is centrally symmetric. Planar central-symmetry conjecture. If every pair of p…

A centrally symmetric polytope is uniquely determined, up to isometry, by its edge-graph, edge lengths, and vertex-origin distances. Central symmetry reconstruction conjecture. The…