Rectangles Inscribed in Locally Connected Plane Continua
Keywords:
hyperspaces, inscribed rectangles, locally connected continua, nth-symmetric product, plane continua, square peg problemAbstract
A plane continuum $X$ is said to admit an inscribed rectangle if for every embedding $\gamma \colon X \to \mathbb{R}^2$, all vertices of at least one Euclidean rectangle lie on $\gamma(X)$. In this paper, we prove that if a plane continuum $X$ contains a copy of the capital letter $\mathsf{H}$ continuum, the simple 4-od, or $S^1$, then $X$ admits an inscribed rectangle. Also, we prove that the only locally connected plane continua that do not admit an inscribed rectangle are the arc and the simple 3-od.
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