  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.
