Mathematical, Computing & Information Sciences
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 is well-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, 2 ≤ k ≤ n − 1, and it allows us to specify a wide variety of desired projection ranges n−k ⊂ Rn−k. Letting n−k be an (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.
Coll, Vincent; Dodd, Jeff; and Harrison, Michael, "The Archimedean Projection Property" (2018). Research, Publications & Creative Work. 43.