Convex solids with planar midsurfaces

Abstract

We show that the boundary of annn-dimensional closed convex setB⊂RnB \subset \mathbb {R}^n, possibly unbounded, is a convex quadric surface if and only if the middle points of every family of parallel chords ofBBlie in a hyperplane. To prove this statement, we show that the boundary ofBBis a convex quadric surface if and only if there is a pointp∈intBp \in \mathrm {int}\,Bsuch that all sections ofbdB\mathrm {bd}\,Bby 2-dimensional planes throughppare convex quadric curves. Generalizations of these statements that involve boundedly polyhedral sets are given.