Convex solids with planar midsurfaces

We show that the boundary of an $n$-dimensional closed convex set $B \subset \mathbb{R}^n$, possibly unbounded, is a convex quadric surface if and only if the middle points of every family of parallel chords of $B$ lie in a hyperplane. To prove this statement, we show that the boundary of $B$ is a convex quadric surface if and only if there is a point $p \in \mathrm{int}\,B$ such that all sections of $\mathrm{bd}\,B$ by 2-dimensional planes through $p$ are convex quadric curves. Generalizations of these statements that involve boundedly polyhedral sets are given.