Variants of the Square Peg Problem

Abstract

[en] The Square Peg Problem, also known as Toeplitz’ Conjecture, is an unsolved problem in the mathematical areas of geometry and topology that states the following: Every simple closed curve in the plane inscribed a square. Even though it seems like an innocent statement, it requires a lot of technical knowledge to proof even when applying certain smoothness conditions to the curve. Over time, variants of this problem have emerged. Some of them offer very interesting results with beautiful proofs. We intend on giving a general historical overview about the Square Peg Problem and the most known variants. Then we will explore the variants related to the inscription of rectangles and triangles and show a few strong results.