The Shadows of a Cycle Cannot All Be Paths

Abstract

A "shadow" of a subset S of Euclidean space is an orthogonal projection of S into one of the coordinate hyperplanes. In this paper we show that it is not possible for all three shadows of a cycle (i.e., a simple closed curve) in R^3 to be paths (i.e., simple open curves). We also show two contrasting results: the three shadows of a path in R^3 can all be cycles (although not all convex) and, for every d 1, there exists a d-sphere embedded in R^d+2 whose d+2 shadows have no holes (i.e., they deformation-retract onto a point).

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