Characterization of spheres by stereographic projection (Q689733)

Let \(S\) be a closed convex surface in \(\mathbb{E}^{d+1}\) with \(c=(0,\ldots,0,1)\in S\) and supporting hyperplane \(\mathbb{E}^ d:x_{d+1}=0\). Assume that for each \(x\in S\backslash\{0\}\) the ray through \(x\) starting at \(c\) meets \(\mathbb{E}^ d\) at a point \(s(x)\), say, and that the mapping \(s(\cdot)\) from \(S\backslash\{0\}\) into \(\mathbb{E}^ d\) thus defined is injective. \(s(\cdot)\) is called stereographic projection of \(S\) into \(\mathbb{E}^ d\). Then \(S\) is a sphere or a truncated sphere if and only if \(s(\cdot)\) is angle-preserving. Here a truncated sphere is the boundary of the intersection of a solid ball in \(\mathbb{E}^{d+1}\) and the halfspace \(x_{d+1}\geq 0\). The proof makes use of tools from measure theory and geometry.