The Archimedean Projection Property

TL;DR

This paper constructs new smooth, compact hypersurfaces in Euclidean space that satisfy generalized Archimedean projection properties for various projection codimensions, extending known results about spheres.

Contribution

It introduces a method to create hypersurfaces satisfying Archimedean projection properties for any projection codimension, broadening the class of such geometric objects.

Findings

Constructed hypersurfaces satisfy projection volume ratio property.

Includes spheres as special cases for codimension 2.

Provides a flexible framework for designing hypersurfaces with specified projection ranges.

Abstract

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.