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We show that the boundary of an n-dimensional closed convex set B ⊂ 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 ∈ int B such that all sections of bd B by 2-dimensional planes through p are convex quadric curves. Generalizations of these statements that involve boundedlydoi:10.1090/s0002-9939-07-09125-3 fatcat:duzxfkzst5b4tdyblp4xpvdq64