A Simple Proof of a Characterization Theorem of the Sphere

S. M. Ulam posed a problem which states: “If a body rests in equilibrium in every direction on a flat horizontal surface, must it be a sphere? ” See [1] for detail. In 1974, L. Montejano [2] solved it as follows. Theorem. If a body D ⊂ R3 rests in equilibrium in every direction, then its convex hull C(D) must be a sphere. (If a planar body D ⊂ R2 rests in equilibrium in every direction, then C(D) must be a circle.) In this note, we give a simple proof of the 2-dimensional case of the above theorem. The reader can prove the 3-dimensional case in the same way. Let D ⊂ R2 be a connected closed region. Then, by the definition, the centroid G of D is an interior point of the convex hull C(D) of D. For every unit vector u⃗, there is a line l orthogonal to u ⃗ such that l ∩ D, ∅ and PQ · u ⃗ 5 0 for every P ∈ l, Q ∈ D. (1) Figure 1 shows the situation. We call l the supporting line of D with respect to the direction u⃗. Let P be the foot of the perpendicular from G to l. We say that D rests in equilibrium in a direction u ⃗ if P ∈ C(D). Figure 2 shows a planar body D which does not rest in equilibrium in the downward direction. l u