Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratios
成果类型：
Article
署名作者：
Hugelmeyer, Cole
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
DOI：
10.4007/annals.2021.194.2.3
发表日期：
2021
页码：
497-508
关键词：
摘要：
We prove that for every smooth Jordan curve gamma, if X is the set of all r is an element of [0, 1] so that there is an inscribed rectangle in gamma of aspect ratio tan(r . pi/4), then the Lebesgue measure of X is at least 1/3. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedded in R x RP3. We prove that any such set of projective planes can be equipped with a natural total ordering. We then combine this total ordering with Kemperman's theorem in S-1 to prove that 1/3 is a sharp lower bound on the probability that a Mobius strip filling the (2, 1)-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.