Loveland, L. D. When a subset of \(E^ n\) locally lies on a sphere. (English) Zbl 0716.57014 Fundam. Math. 133, No. 2, 101-112 (1989). Let X be a continuum in 2-space such that each point p of X is the vertex of two triangles that are symmetric about p, have angles greater than 60\(\circ\) at p, and whose interiors miss X. This paper shows that X must be an arc or a simple closed curve. A similar theorem is proved in 3- space. If X is a finite polyhedron in 3-space with no local cut points such that each point of X is the vertex of two cones that are symmetric about p, have angles greater than 60\(\circ\) at p, and whose interiors miss X, then X locally lies on a 2-sphere. This paper does not depend on any of its references and briefly discusses the problem in n-space, \(n>3\). Reviewer: D.G.Wright MSC: 57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010) 57N35 Embeddings and immersions in topological manifolds 52B10 Three-dimensional polytopes Keywords:embedding; finite polyhedron in 3-space with no local cut points PDFBibTeX XMLCite \textit{L. D. Loveland}, Fundam. Math. 133, No. 2, 101--112 (1989; Zbl 0716.57014) Full Text: DOI EuDML