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Rectangles inscribed in plane sets as a consequence of the non-embeddability of certain cones in R3

Ulises Morales-Fuentes ⟨ulises.morales@uaem.mx⟩

Abstract:

A plane set admits an inscribed rectangle if every homeomorphic copy of it in R2 contains the 4 vertices of at least one Euclidean rectangle. Vaughan proved that S1 admits an inscribed rectangle by reducing the problem to the non-embeddability of the projective plane in R3 (it is not known if S1 admits an inscribed rectangle of aspect ratio 1:1 i.e. a square). In this talk, using the non-embeddability of the Cone(K5) and the Cone(K3,3) in R3 we classify plane compact connected locally-connected sets that admit inscribed rectangles. Using similar topological techniques, we also present a one-dimensional non-connected set such that every copy of it in R2 admits an inscribed rectangle with at least one vertex in each component.

Status: Accepted

Collection: Geometric Topology

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