Characterization of the sphere and of bodies of revolution by means of Larman points

Abstract Let n ≥ 3 and let K ⊂ ℝ n be a convex body. A point p ∈ int K is said to be a Larman point of K if for every hyperplane Π passing through p , the section Π ∩ K has an ( n – 2)-plane of symmetry. If p is a Larman point of K and for every section Π ∩ K , p is in the corresponding ( n – 2)-plane of symmetry, then we call p a revolution point of K . We conjecture that if K contains a Larman point which is not a revolution point, then K is either an ellipsoid or a body of revolution. This generalizes a conjecture of Bezdek for n = 3. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if K ⊂ ℝ n is a strictly convex origin symmetric body that contains a revolution point p which is not the origin, then K is a body of revolution. This generalizes the False Axis of Revolution Theorem in 7. We also show that if p is a Larman point of K ⊂ ℝ 3 and there exists a line L such that p ∉ L and, for every plane Π passing through p , the line of symmetry of the section Π ∩ K intersects L , then K is a body of revolution (in some cases, K is a sphere). We obtain a similar result for projections of K . Additionally, for K ⊂ ℝ n with n ≥ 4, we show that if every hyperplane section or projection of K is a body of revolution and K has a unique diameter D , then K is a body of revolution with axis D .