The Archimedean projection property
-
Vincent Coll
,
Jeff Dodd
Abstract
Let H be a hypersurface in ℝn and let π be an orthogonal projection in ℝn restricted to H. We say that H satisfies the Archimedean projection property corresponding to π if there exists a constant C such that Vol(π–1(U)) = C·Vol(U) for every measurable U in the range of π. It iswell-known that the (n–1)-dimensional sphere, as a hypersurface in ℝn, satisfies the Archimedean projection property corresponding to any codimension 2 orthogonal projection in ℝn, the range of any such projection being an (n – 2)-dimensional ball. Here we construct new hypersurfaces that satisfy Archimedean projection properties. Our construction works for any projection codimension k with 2 ≤ k ≤ n – 1, and it allows us to specify a wide variety of desired projection ranges Ωn–k ⊂ ℝn–k. Letting Ωn–k be the (n – k)-dimensional ball for each k, it produces a new family of smooth, compact hypersurfaces in ℝn satisfying codimension k Archimedean projection properties that includes, in the special case k = 2, the (n – 1)-dimensional spheres.
Acknowledgements
We thank Jerry King for bringing Bernstein’s theory to our attention, David L. Johnson, Joe Yukich, and Rob Neel for reading and commenting on various drafts of this paper, and an anonymous referee for helpful suggestions that substantially improved the final product.
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© 2017 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Frontmatter
- Classification of 3-dimensional left-invariant almost paracontact metric structures
- On the topology of the spaces of curvature constrained plane curves
- On deformations of parallel G2 structures and almost contact metric structures
- A flag representation of projection functions
- A fundamental theorem for submanifolds of multiproducts of real space forms
- Intriguing sets of quadrics in PG(5, q)
- Successive radii and ball operators in generalized Minkowski spaces
- Optimal inequalities for the normalized δ-Casorati curvatures of submanifolds in Kenmotsu space forms
- The Archimedean projection property
- Kernels of numerical pushforwards
- Modified classical flat Minkowski planes
Articles in the same Issue
- Frontmatter
- Classification of 3-dimensional left-invariant almost paracontact metric structures
- On the topology of the spaces of curvature constrained plane curves
- On deformations of parallel G2 structures and almost contact metric structures
- A flag representation of projection functions
- A fundamental theorem for submanifolds of multiproducts of real space forms
- Intriguing sets of quadrics in PG(5, q)
- Successive radii and ball operators in generalized Minkowski spaces
- Optimal inequalities for the normalized δ-Casorati curvatures of submanifolds in Kenmotsu space forms
- The Archimedean projection property
- Kernels of numerical pushforwards
- Modified classical flat Minkowski planes
