The Archimedean projection property

Abstract

Abstract Let H be a hypersurface in ℝ n and let π be an orthogonal projection in ℝ n 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 ℝ n , satisfies the Archimedean projection property corresponding to any codimension 2 orthogonal projection in ℝ n , 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 Ω n–k ⊂ ℝ n–k . Letting Ω n–k be the (n – k)-dimensional ball for each k, it produces a new family of smooth, compact hypersurfaces in ℝ n satisfying codimension k Archimedean projection properties that includes, in the special case k = 2, the (n – 1)-dimensional spheres.