The Archimedean Projection Property

Let H be a hypersurface in R-n and let pi be an orthogonal projection in R-n restricted to H. We say that H satisfies the Archimedean projection property corresponding to p if there exists a constant C such that Vol(pi(-1)(U))= C.Vol(U) for every measurable U in the range of pi. 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 Omega(n-k) subset of Rn-k. Letting Omega(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.