2019/11/17 by Cole Hugelmeyer, Hugelmeyer, Cole
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Metric Geometry (math.MG) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1911.07336
openalex publication_date 2019/11/17 · openalex created_date 2022/07/22 · openalex updated_date 2026/07/28
We prove that for every smooth Jordan curve \γ, if X is the set of\nall r \∈ [0,1] so that there is an inscribed rectangle in \γ of aspect\nratio \tan(r\⋅ \π/4), then the Lebesgue measure of X is at least 1/3.\nTo do this, we study sets of disjoint homologically nontrivial projective\nplanes smoothly embedded in \ℝ\× \ℝP3. We prove that any\nsuch set of projective planes can be equipped with a natural total ordering. We\nthen combine this total ordering with Kemperman's theorem in S1 to prove\nthat 1/3 is a sharp lower bound on the probability that a M "obius strip\nfilling the (2,1)-torus knot in the solid torus times an interval will\nintersect its rotation by a uniformly random angle.\n