The Archimedean Projection Property

AbstractLet H be a hypersurface in R n and let π be an orthogonal projection in R n restrictedto H. We say that H satisfies the Archimedean projection property correspondingto π if there exists a constant C such that Vol(π −1 (U)) = C·Vol(U) for every mea-surable U in the range of π. It is well-known that the (n−1)-dimensional sphere, asa hypersurface in R n , satisfies the Archimedean projection property correspondingto any codimension 2 orthogonal projection in R n , the range of any such projectionbeing an (n − 2)-dimensional ball. Here we construct new hypersurfaces that sat-isfy Archimedean projection properties. Our construction works for any projectioncodimension k, 2 ≤ k ≤ n− 1, and it allows us to specify a wide variety of desiredprojection ranges Ω n−k ⊂ R n−k . Letting Ω n−k be an (n − k)-dimensional ball foreach k, it produces a new family of smooth, compact hypersurfaces in R n satisfyingcodimension k Archimedean projection properties that includes, in the special casek = 2, the (n−1)-dimensional spheres.