The Archimedean Projection Property

Let $H$ be a hypersurface in $\mathbb R^n$ and let $\pi$ be an orthogonal projection in $\mathbb R^n$ restricted to $H$. We say that $H$ satisfies the $Archimedean$ $projection$ $property$ corresponding to $\pi$ if there exists a constant $C$ such that $Vol(\pi^{-1}(U)) = C \cdot Vol(U)$ for every measurable $U$ in the range of $\pi$. It is well-known that the $(n-1)$-dimensional sphere, as a hypersurface in $\mathbb R^n$, satisfies the Archimedean projection property corresponding to any codimension 2 orthogonal projection in $\mathbb R^n$, 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 $k$, $2 \leq k \leq n - 1$, and it allows us to specify a wide variety of desired projection ranges $\Omega^{n-k} \subset \mathbb R^{n-k}$. Letting $\Omega^{n-k}$ be an $(n-k)$-dimensional ball for each $k$, it produces a new family of smooth, compact hypersurfaces in $\mathbb R^n$ satisfying codimension $k$ Archimedean projection properties that includes, in the special case $k = 2$, the $(n-1)$-dimensional spheres.