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When a subset of \(E^ n\) locally lies on a sphere. (English) Zbl 0716.57014

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
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