2023/07/18 by Alfonseca, María Angeles, Cordier, Michelle, Jerónimo-Castro, Jesús +1 · 1 citation
#FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2307.09585
Let K⊂ ℝn, n≥ 3, be a convex body. A point p the interior of K is said to be a Larman point of K if for every hyperplane Π passing through p the section Π∩ K has a (n-2)-plane of symmetry. If p is a Larman point of K and, in addition, 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 K. Bezdek for convex bodies in ℝ3 to n ≥ 4. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if K ⊂ ℝn 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. We also show that if p is a Larman point of K ⊂ ℝ3 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, we conclude that K is a sphere). We obtain a similar result for projections of K. Additionally, for K ⊂ ℝn, 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.