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arXiv · 1507.02355

The Shadows of a Cycle Cannot All Be Paths

Prosenjit Bose·Jean-Lou De Carufel·Michael G. Dobbins·Heuna Kim·Giovanni Viglietta
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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 $\mathbb R^3$ to be paths (i.e., simple open curves). We also show two contrasting results: the three shadows of a path in $\mathbb R^3$ can all be cycles (although not all convex) and, for every $d\geq 1$, there exists a $d$-sphere embedded in $\mathbb R^{d+2}$ whose $d+2$ shadows have no holes (i.e., they deformation-retract onto a point).

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Prosenjit Bose, Jean-Lou De Carufel, Michael G. Dobbins, Heuna Kim, Giovanni Viglietta. 2015-07-09. The Shadows of a Cycle Cannot All Be Paths. https://arxiv.org/abs/1507.02355

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