# Proving that a solid is a ball if all plane sections of it are circles

All plane sections of a solid are circles. Prove that the solid is a ball.

Got this from Arthur Engel’s book. The incomplete solution given was as follows;

Consider the largest chord of the solid. Any section through this cord is a circle whose diameter is the chord. Otherwise the circle and the solid would have a larger chord. Thus the solid is a ball and one of its diameters is the selected chord.

Engels does admit that the proof is not complete, citing that the proof did not address whether a longer chord exists, and that

In fact, if the surface of the solid did not belong to the solid, a longer chord would not exist. So we assume that the solid is a closed and bounded set. Then we can apply the theorem of Weierstraβ: A continuous function defined on a closed and bounded set always assumes its global maximum and minimum.

He finally closes by stating that “There are also elementary proofs which are slightly longer (see HMO 1954)”.

My query is whether anyone is able to find elementary proofs to this question, as well as to explain Engel’s solution.

I suspect it can be easily(?) resolved with the use of integration and Riemann sums, but I would much prefer a purer proof of the question.