Characterization of the sphere and of bodies of revolution by means of Larman points
-
M. Angeles Alfonseca
,
M. Cordier
Abstract
Let n ≥ 3 and let K ⊂ ℝn 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 ⊂ 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 in [7]. 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, K is a sphere). We obtain a similar result for projections of K. Additionally, for K ⊂ ℝn 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.
Funding statement: The first author is supported in part by the Simons Foundation gift 711907. The fourth author is supported by the National Council of Sciences and Technology of Mexico (CONACyT) Grant I0110/62/10 and SNI 21120.
Acknowledgements
This work was partially done during a sabbatical year of the fourth author at University College London (UCL). The fourth author thanks UCL for their hospitality and support. We thank Dmitry Ryabogin for many fruitful discussions about these results. We also thank the referee for many valuable suggestions to improve the clarity of the paper.
Communicated by: M. Henk
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Articles in the same Issue
- Frontmatter
- Inequalities for f*-vectors of lattice polytopes
- Lower bound on the translative covering density of octahedra
- Variations on the Weak Bounded Negativity Conjecture
- Poisson Structures on moduli spaces of Higgs bundles over stacky curves
- Generalized Shioda–Inose structures of order 3
- Deformation cones of Tesler polytopes
- Some observations on conformal symmetries of G2-structures
- Characterization of the sphere and of bodies of revolution by means of Larman points
- Fractional-linear integrals of geodesic flows on surfaces and Nakai’s geodesic 4-webs
- The feet of orthogonal Buekenhout–Metz unitals
Articles in the same Issue
- Frontmatter
- Inequalities for f*-vectors of lattice polytopes
- Lower bound on the translative covering density of octahedra
- Variations on the Weak Bounded Negativity Conjecture
- Poisson Structures on moduli spaces of Higgs bundles over stacky curves
- Generalized Shioda–Inose structures of order 3
- Deformation cones of Tesler polytopes
- Some observations on conformal symmetries of G2-structures
- Characterization of the sphere and of bodies of revolution by means of Larman points
- Fractional-linear integrals of geodesic flows on surfaces and Nakai’s geodesic 4-webs
- The feet of orthogonal Buekenhout–Metz unitals
