Rectangles Inscribed in Locally Connected Plane Continua - Morales-Fuentes, Ulises; Villanueva-Segovia, Cristina (Topology Proceedings, 2020)
A plane continuum $X$ is said to admit an inscribed rectangle if for every embedding $γ \colon X → \mathbb{R}^2$, all vertices of at least one Euclidean rectangle lie on $γ(X)$. 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.